1. The Gemora (Yevomos 91A) says when the wife gives the husband Maase Yodayim it is a transfer of money from the woman to he husband. Tosfos (D'H Kol Sheken) asks, but if she is giving Maasei Yodayim, he would be giving her Mezonos. So then it is not just from her to him but also transfer of money from him to her. So Tosfos answers that most women earn more Maasei Yodayim that they eat in Mezonos. So the net flow is from her to him.
However we know that the whole reason of the Takona of Chazal in the Kesuba to provide a woman with Mezonos was that generally women could not earn enough Maasei Yodayim to cover her Mezonos. In the exceptional case where the woman was able to earn more, she could choose to say "Eini NizunisVe'eini Ose". Isn't our Tosfos in conflict with this?
2. The Gemora (Yevomos 91B) says that a woman who has remarried based on a Get that was Posul, is considered negligent in that she should have read and checked the Get. Rashi explains that she should have shown it to a Chochom. Even if she does not know what to look for, (most men would not know either), a Chochom who is Yodeia betiv Gittin UKiddushin would know.
We see from this Gemora that a Get was given in those days without the involvement of Bes Din. The husband would just go to a Sofer, have the Get written up and give it to his wife. Surely if given in Bes Din, the role of Bes Din would be to determine that the Get was 100% Kosher and she would not be expected to show the Get to an independent Chochom.
As far as I know today, Gittin are only given in Bes Din. When did this practice begin? Was the reason for this exactly our case, to stop people giving Gittin that contained errors?
Yisroel Alter Pacanowski
(1) I found that the Yashresh Yaakov here also understands this matter in a similar way to you: that it is frequent that a woman does not earn sufficient to cover her food expenses, and in fact he writes that this is what the Gemara (above 90b) means when it states "What is the reason that Rabanan said that the husband receives Ma'aseh Yadayim? Because she eats Mezonos". However Yashresh Yaakov also does not give an answer to his question on Tosfos. Therefore I would like to show an additional source that most women do earn their way.
This is from Rashi in Kesuvos 58b DH Mezoni. There the Gemara cites Rav Huna's Din that the wife can say that she will not take food from her husband and will not give him her earnings. The Gemara then suggests that Rav Huna maintains that the principal institution was that the husband must feed his wife and as a result of this, Chazal said that he receives her earnings to prevent ill-feelings. Rashi explains that the original enactment was for her benefit, because "Zimnin" her earnings are not sufficient.
The crucial question is: what exactly does the word "Zimnin" mean? See Taz YD 251:1 that it means "on one occasion". (See Chidushei Ha'Gershuni there on the margin of the Shulchan Aruch who cites a few examples of this.) I would like to offer another source that Zimnin means something rare: from the Gemara below 109a which states that people do not always know when a husband has taken his wife back. Rashi DH Beki'im writes that "Zimnin" a man re-instates his wife in the evening and dies the next morning. This must be a fairly rare scenario, so we have another proof that Zimnin means something unusual.
Therefore we see from Rashi Kesuvos 58b that it is not usual that a woman does not earn enough to cover her food bill. Nevertheless Chazal were still concerned about the minority that do not, and made a Din that that the husband is obligated to feed his wife. So one does not have to say like you wrote that it is only in an expectional case where she earns more, but rather we can say that even though she does usually earn more, nevertheless most husbands are generous to their wives and allow them to enjoy the money that they earn. In a case where the wife feels her husband is not being fair to her, she has the right to say "Eini Nizunis ve'Eini Ose".
(2) Abaye does indeed exclaim, in Erchin 29a and Bava Basra 174b, "does everyone who divorces, do so in Beis Din?!", which suggests that a Get does not have to be given in Beis Din.
However, the question of whether or not a Get is considered a Din is not a simple one. The Bartenura Mishnah Bechoros 4:6 criticizes Rabbis who receive payment - more than they could have earned if they were doing some different work - for dealing with the Get, in contradiction to the Mishnah that
if a Dayan gets paid for judging a Din Torah, the case is thereby disqualified. The Rema EH (The Order of the Get 154:3) disagrees with Bartenura and writes that arranging a Get is not considered a Din. The Pischei Teshuvah #6 asks on the Rema from the Halachah that one must not give a Get at night, which one has to say is because a Get is considered a Din Torah, which is invalid at night.
The Pischei Teshuvah cites the Noda b'Yehudah who answers that one has to distinguish between different aspects of the Get. The giving of the Get from the hand of the husband to his wife is indeed considered a Din and must not be done at night, and must be performed in Beis Din. In contrast the stages preliminary to this - the writing of the Get with all its details is not considered a Din (therefore the Rabbi is allowed to receive pay to teach how to do it).
[See also Noda b'Yehudah, in full, end of DH v'Hinei Ma'aloso, who explains the Gemara Erchin 29a that one does not require a Beis Din of 3 experts, but one does require the get to be given in front of 3 people].
Therefore it seems that one can explain that in our Gemara, since a Beis Din is not strictly required for the preparing of the Get, it is possible that a mistake was made, and therefore the wife is considered negligent for not showing it to a Chacham. However a Beis Din of 3 was always required for the actual handing over of the Get from husband to wife but it is not actually essential according to the Din of the Gemara that experts should read it at this stage. Presumably nowadays, because it is more likely that mistakes will be made, it is customary to require a qualified Beis Din to deal with every stage of the Get.
Here is a different explanation why Chazal gave the wife Mezonos even though most women earn more than they eat. The reason is simply because this way is more convenient for the ladies.
After the Takanah of Chazal, the husband has to see to it that he always supplies his wife with sufficient food (or alternatively gives her money regularly to buy her food). This way, whatever the situation is with her earnings (e.g. at times of life when it may be difficult for her to work) the wife can feel calm that her husband will always look after her.
If she feels that she is not being looked after properly, she can rely on what Rav Huna states (Kesuvos 58b) that she can say "I will not take food from you, and will not give you my earnings", but most women will presumably not want to do this, bur rather will rely on their husbands to look after them.
Kesivah u'Chasimah Tovah
From the following mishna it would appear clear that women often produced more than what she eats-
KETUBOT: CHAPTER 5- MISHNAH 4
If one consecrates his wife's handiwork, she works and eats. The surplus - Rabbi Meir says, It is consecrated. Rabbi Yohanan HaSandlar says, It is hullin.
Yeshayahu HaKohen Hollander
I do not see that there is necessarily a proof from the above Mishnah alone that most women produce more than what they eat, because everyone would agree that sometimes they overproduce (i.e. even the Yashresh Yaakov that I cited who maintains that most women earn less than their food requirements) and it is quite possible that the Mishnah is discussing the minority of women (as we know the Gemara often mentions quite unusual scenarios). However the problem was since there is a rule that Chazal did not make decrees for unusual cicumstances (see Beitzah end 2b and frequently) it is logical that Chazal would not make a Takanah that the food would go to the women in replace for their earnings, if the majority of women did not require this because they earned more than their food needs.
However there may be a proof from the Sugya in the Gemara on the Mishnah that you referred us to, because there - in Kesuvos 59a - the Gemara discusses why Chazal instituted that the husband must give his wife a "Ma'ah Kesef"- a silver coin for additional needs. Rav and Shmuel maintain that they instituted this coin in return for the fact that the husband receives her surplus earnings, more than she needs for food. If it would not be frequent that a woman would earn more than the minimum amount, it is unlikely that Chazal would have bothered to make such a Takanah, so one can offer a proof from here for Tosfos Yevamos 91a that most women produce more than what they eat.
Kesivah u'Chasimah Tovah
Here is a different answer to the question about the Get outside of Beis Din.
(1) One can say that in fact all aspects of the Get are valid b'Di'eved even if done outside of Beis Din. This fits in with the simple meaning of Bava Basra 174b that I cited in my first reply.
(2) However we saw that some say that a Get must not be given at night. In fact the Rema in Shulchan Aruch EH end 123 cites some opinions that it is invalid even b'Di'eved if given at night and some explain that the reason for this is because a Get is a Din (see Chelkas Mechokek there #13 and Beis Shmuel #9) which cannot be done at night. If a Get is a Din, this suggests it needs a Beis Din.
However we can now reject this argument and say that in fact a Get is considered a Din even though it was not given in Beis Din. This can be explained further with the help of the Teshuvas Or Zarua #745 (by one of the Rishonim), printed in part 1 of the Or Zarua, near the end of p.211. The Or Zarua maintains at length that a Get is invalid if given at night. He asks that according to this, one should be required to give the Get over in front of 3 people, so why do we never find a standard case in the Gemara that one must hand it over in front of 3? He answers that the word "Mishpat" (judgement") is stated by the Torah in connection with Get (see Gitin 88b where the verse in Shemos 21:1 - "And these are the judgements" - is stated in connection with a Get). If so this should be similar to what R. Yochanan said in the Gemara (Yevamos 46b) that since "Mishpat" is stated in connection with accepting converts, it follows that one requires 3 people to do the latter. However the Or Zarua writes that R. Yochanan's Din is only mid'Rabanan. Even in monetary matters the Halachah is that one Dayan is valid mid'Oraisa (see Sanhedrin 3a) and certainly for conversions one is valid mid'Oraisa. For a Get, Chazal did not require 3 even mid'Rabanan.
(3) In fact the Maharam Shif, at the beginning of Sanhedrin 2a, writes that one does not require a Beis Din for Get and since we see that the Gemara in Gitin 9b (and in several other places) states that according to R. Eliezer it is the witnesses who observe the Get being handed over who make the Get valid, while according to R. Meir the witnesses signed on the Get are the crucial factor (see Gitin, top 3b), one sees at any rate that the witnesses make the Get. Therefore one only requires two, not three (i.e. a Beis Din is not crucial).
(4) Pischei Teshuva (Shulchan Aruch EH 154:8, in "Order of the Get") also writes that it is not stated anywhere in Shulchan Aruch and Acharonim that one needs Beis Din for a Get. He cites authorities who disagree with the Noda b'Yehudah I mentioned in my previous reply that one must have 3 for the handing over of the Get. The Rashash on Sanhedrin 2a DH Mi'unin also disagrees with the Noda b'Yehudah.
(5) However Rabeinu Gershom, in his commentary to Bava Basra 174b, seems at first sight to imply that one must have a Beis Din for a Get and to support the Noda b'Yehudah. On the Gemara there that we cited above (1), R. Gershom explains "Does everyone who divorces, do so in a Great Beis Din - even in a Beis Din of laymen - who do not know the Din of the Mishnah there 173b that the husband divorcing his wife should make a vow that she cannot derive benefit from him (see the Sugya there for the details) - one may divorce." This explanation of R. Gershom suggests that while not everyone divorces in an expert Beis Din, nevertheless
one must divorce at least in front of 3 ordinary citizens.
However this proof from R. Gershom can be refuted. It may be that even R. Gershom agrees that it is only l'Chatchilah that one must give a Get even in front of 3 laymen. When he writes "Even in a Beis Din of laymen one can divorce", this means that even l'Chatchilah one can divorce in front of a Beis Din of laymen but b'Di'eved a Get without a Beis Din at all is also valid.
(6) The above Pischei Teshuva cites the Noda b'Yehudah who maintains strongly that a Beis Din is required for a Get. He also cites Targum Yonasan on Devarim 24:1 that the husband writes the Get in front of Beis Din. The Urim v'Tumim (by R. Yonasan Eibeshitz) writes that based on the Targum Yonasan the custom became widespread to require a Beis Din for a Get.
The Chaim Sha'al (by the "Chida" - R. Chaim Yosef David Azulai) writes that in most Jewish communities the Get is given in front of a Beis Din of 3. The reason for the custom is because there are a lot of intricate Halachos concerning Gitin so one requires experts. In some places the custom was even to gather together most of the scholars of the city before a Get was given. The Chazon Ish EH 103:42 also writes that one always requires a Beis Din because otherwise she will not possess a Chazakah of being divorced.