Why does the Torah insert the word "Zeh", which generally implies something visible?
Rashi and Targum Yonasan: Because Hashem showed him a coin of fire 1 to demonstrate exactly how the half-Shekel should look.
Hadar Zekenim: Because, since they lowered themselves [at the Eigel] with the word "Zeh" - when they said "Zeh Elohecha" (Nechemyah, 9:18), or "Ki Zeh Moshe ha'Ish asher He'elanu" (32:1) thereby attributing Yetzi'as Mitzrayim to Moshe, and not to Hashem), they now elevated themselves with "Zeh Yitnu". 2
Yerushalmi Shekalim, 6:3: It teaches us that any Kohen who does not donate a hslf-Shekel is a sinner - by virtue of the fact that the Gematriya of "Zeh" is twelve, thereby including all the tribes. 3
Da'as Zekenim, Hadar Zekenim (12, citing Tanchuma): He showed him with His right thumb. All the fingers of His right hand are connected with Ge'ulah. Also see Ba'al ha'Turim.
Hadar Zekeinim: As the Pasuk says in Tehilim 75:8 "Zeh Yashpil ve'Zeh Yarim".
However this is only an Asmachta. See Torah Temimah, who elaborates in detail.
Why does the Torah use the expression "Kol ha'Over al ha'Pekudim"?
Rashi: When one counts, it is customary to pass whoever (whatever) is being counted before the one who is counting. 1
As we find in Bechukosai, Vayikra 27:32 and in Yirmiyah, 33:13. See also K'li Yakar.
Why was specifically a half-Shekel required to atone for their sin?
Yerushalmi Shekalim, 2:3 #1 (citing R; Yehudah or R. Nechemyah): Because they sinned at midday (when Moshe did not return - Shabbos 89a).
Yerushalmi Shekalim, 2:3 #2 (citing R; Yehudah or R. Nechemyah): It is because they sinned after six hours - 'Let them give half a Shekel, which is equal to six Garmisin'. 1
Yerushalmi Shekalim, 2:3 #3 (citing R. Nechemyah in the name of R. Yochanan ben Zakai): Because they transgressed the Ten Commandments (and half a Shekel equals ten Geirah). 2
The Gematriya of Shekel is 'Nefesh'. Hashem in His mercy atones for the sins of half the Nefesh, so a half-Shekel atoned for the other half.
Oznayim la'Torah: It is physically impossible for man to build a dwelling for Hashem who created Heaven and earth) and to bring sacrifices to a G-d who does not eat. 3 Therefore Hashem told Moshe that Yisrael should do what they can (half) and He will supply the other half - by coming to dwell in the Mishkan that they build and consuming the Korbanos that they bring (the other half).
What is a Geirah?
See Shmuel 1, 2:36. Initially, a Zuz comprised five Ma'ah, but Chazal added one sixth (a fifth of the initial value), to make it six (Rashi). See also Ramban (on Pasuk 12), who cites a Pasuk in Nechemyah that they gave a third of a Shekel (ten Geirah) - because Ezra raised the value of the Shekel.
And one Shekel equals four Zuzim (Rashi).
What is the significance of the term "Shekel ha'Kodesh"?
Rashi: It is the Shekel that Hashem fixed to use for Divrei Kedushah, such as the Shekalim mentioned in Bechukosai (chapter 27) in connection with Erchin and Sadeh Achuzah.
Ramban #1: Whatever emanates from the Torah bears the title 'Kodesh'. Consequently, the language of the Torah, with which Hashem communicated with the prophets and with Yisrael 1 is called Lashon ha'Kodesh, And by the same token, all sums of money mentioned in the Torah (which are all in terms of Shekalim) are called Kodesh.
Ramban #2 (citing Rambam in Moreh Nevuchim): The language is called Lashon ha'Kodesh because there are no intrinsic words for the Eivarei Tashmish of a man or a woman, nor for a drop of Keri, urine or excrement - they are all presented with descriptive titles which convey their meaning. 2
Ramban: Moreover it is the language by which Hashem Himself is named (Keil, Elokim ... and Havayah), and it is the language which He used to create the world, and with which He named Heaven and earth and all their hosts, as well as all the angels (Micha'el, Gavri'el).
See Ramban's objection to this explanation.
What is the value of the Shekel mentioned in the Torah?
Ramban: The Shekel mentioned in the Torah, 1 which weighs four gold coins 2 , is actually the equivalent of a Sela (comprising four Dinarim). 3
What is the significance of the three 'T'rumos mentioned in Pesukim 13, 14 & 15?
Megilah, 29b: One for the Mizbe'ach (the Korbanos), one for the sockets and one for Bedek ha'Bayis. 1
Why are there two notes (Kadma Ve'azla) on the word "Venasnu"?
Kol Eliyahu and Divrei Eliyahu: Because poverty is a wheel that revolves in the world. One should give Tzedakah (early) before 1 his children will need to receive, and then they will receive promptly (Shabbos 151b). "Venasnu" reads the same backwards and forwards.
Kadma Ve'azla means to go early (PF)
QUESTIONS ON RASHI
Rashi writes that Hashem showed Moshe a 'coin of fire.' Why was this necessary?
Hadar Zekenim (in Pasuk 12) and Moshav Zekenim (25:8): Because one of three 1 things that puzzled Moshe was how such a small donation could attain atonement for the soul - until Hashem showed him a coin of fire). 2
Hadar Zekenim and Moshav Zekenim: The others were a. how Hashem, whom Shamayim does not contain, could dwell in the Mishkan, and b. why He commanded only two lambs a day for the Tamid.
Michtav me'Eliyahu (5, p. 417): Becuse if it was a matter of quantity, there is no limit as to how much is needed to redeem oneself from sin. Therefore, Hashem took a Shekel of fire from under the Kisei ha'Kavod (Tanchuma 7), to hint that where there is pure zeal for Hashem's honor, amidst Hashem's Chesed, even a little will suffices - as the Midrash states in Shir Hashirim Rbah, 5:3 "Open for Me like a needle's eye ... ". (This teaches us that it is not so much the giving that atones, but the way in which one gives it).
