1) Amud a: Elyahu told Rav Yossi "What you heard , not only at this moment,
but every day three times a day, these words are said". Elyahu was
referring to the meimra of Rav Ytschok bar Shmuel in the name of Rav (
which preceeds the story of Rav Yossi) which states that the words which
Rav Yossi heard are those which" At every mishmor Hash-m exclaimed like a
lion". If so, then Rav Yossi could have resolved the Gemara's question
earlier : When does the sign of each mishmar occur, at the beginning , at
the end or at the middle ( for the second mishmor) ? The gemara tries two
possibilities, without a firm conclusion. Yet, since Rav Yossi heard those
words then he could have possibly determine at what part of which mishmor
he heard the voice.
2) Amud b: The Gemara asks: Is it possible that King David knew when
mid-night takes place while Moshe Rabenu didn't know ?
Why didn't the Gemara answer that according to the first explanation for
the division of the night according to Rabbi Eliezer ( amud a) at mid-night
the dogs bark, therefore King David knew when was mid-night from the
barking of the dogs. However, Moshe Rabenu could not tell on the night of
Makot Bechorot when mid- night was because that particular night the dogs
didn't bark "Lo yecheratz kelev leshono" ?
1) Excellent question! It could be that indeed Rebbi Yosi knew when the signs of each Mishmar occur. However, he did not suggest an answer because he was a Tana who lived before the Amora'im who asked the question. To Rebbi Yosi, there never was a question, so he never bothered answering it.
However, it is indeed difficult to understand why Rebbi Yosi did not resolve another question -- the debate between Rebbi Nasan and Rebbi (on Daf 3b). They argued whether there are 3 or 4 Mishmaros. Since Rebbi Yosi lived in their generation, it is likely that he knew of their argument, and what he heard from Eliyahu should have enabled him to resolve their debate. It could be that what Eliyahu told him was not referring back to the statement of Rav Yitzchak Bar Shmuel in the name of Rav. Eliyahu did not mention three Mishmaros, but only "three times every day." And even if Eliyahu was referring to the three Mishmaros, Rebbi Yosi had no way to prove it that it was, so he did not want to use what he heard as a proof.
2) (a) The verse says that the dogs did not bark "at all of the people of Israel" (Shemos 11:7). Only at the Jews did the dogs not bark, but they did bark at the Egyptians (or at cars), and thus Moshe did have a way of knowing when midnight was.
(b) The verse that says that the dogs did not bark at the Jews refers to the moment at which they were leaving Egypt -- that is, in the morning (see Daf 9a). At midnight a few hours before, though, they could have barked.
(c) Moshe was standing and speaking on the night after the thirteenth of Nisan, speaking about the following night. How did he know that the following night the dogs would not bark and he would not know the exact moment of midnight?
The approaches that you suggest to answering your question all seem to be valid ways of understanding the Gemara.