Why does the Pasuk employ the double exclusion "ha'Rak Ach"?
Rashi: Miriam and Aharon were saying was "Did Hashem speak only to Moshe (and not to anybody else)?" 1
Moshav Zekenim: Did Hashem not speak also with the Avos, and they did not separate from Peru u'Rvu?! Moshe separated for his own [desire, without necessity].
See Sifsei Chachamim.
What exactly was their criticism of Moshe?
Rashi: They complained that Hashem spoke with them too, yet they did not separate from their spouses! Moshav Zekenim (1, citing R. Yehudah ha'Chasid)
Why does the Torah add a 'Beis' to the word "Moshe"?
Rashi: As in that Pasuk and here in Pasuk 8.
What is the significance of the phrase "Vayishma Hashem"?
Seforno: It teaches us that Hashem will always take the side of a Talmid-Chacham. 1
Refer to 12:1:151:1.
Seforno: As the Gemara states in B'rachos, 19a 'A Talmid-Chacham is different inasmuch as Hashem defends his Kavod!' See also Ba'al ha'Turim.
Rashi writes that they complained that Moshe should not have separated from his wife. Hashem told him to! "Emor Lahem Shuvu Lachem l'Ohaleichem v'Atah Po Amod Imadi" (Devarim 5:27-28)!
Moshav Zekenim (Shemos 19:15): They understood that Moshe separated before Hashem told him; later, Hashem told him to do so, for in the way one wants to go, Shamayim leads him (Makos 10b). He did not tell Aharon to separate! Moshe told them his Kal v'Chomer (Bnei Yisrael had to separate before Hashem spoke to them) and made a separate tent for himself, or he gave a Get. They rejected the Kal v'Chomer
Rashi writes that they complained that Moshe should not have separated from his wife. Why did they complain only now?
Da'as Zekenim, Moshav Zekenim (Shemos 19:15): From Tziporah's words, they knew that he separated.
Rashi (Devarim 5:28) writes that "Amod Imadi" obligated Moshe to separate from his wife, Shabbos 87a says that Moshe decided to separate from his wife due to a Kal v'Chomer (from Bnei Yisrael, who needed to separate)!
Riva (Shemos 19:15): Once Hashem commanded Yisrael to separate, Moshe divorced his wife. Aharon and Miryam said that had he not done so, Hashem would not have commanded him, for Moshe had prophecy a long time before this, and Hashem did not make him separate! 1
Riva: Even though Tziporah said 'from when Hashem spoke to him, he separated from me', that was a temporary separation. He divorced her only after Bnei Yisrael needed to separate.