Why did the ass take Bil'am to task only after the third time?
Rashi: Refer to 22:26:1:1.
Na'ar Yonasan: Because the first two times Bil'am had reason to strike it - for straying from the path; whereas the third time he did not. 1
Na'ar Yonasan. Note also that the third time Bil'am struck it with a staff, which evidently he did not do the first two times.
Why did Hashem cause the ass to speak?
Seforno: To teach Bil'am that Hashem is the Master of speech - and if He can make an ass speak, He can certainly prevent him from speaking, should He so wish. 1
How does the ass speaking conform to the principle that, since the time of the creation, Hashem never created anything new?
Targum Yonasan, Avos 5:6: The mouth of the ass is one of the ten miraculous creations that Hashem created Erev Shabbos at dusk. 1
Targum Yonasan: The others are the Manna; Miriam's well; Moshe's staff; the Shamir ('worm'); the rainbow; the Ananei Kavod; the mouth of the earth (of Korach),; the writing on the Luchos; and the Mazikin (demons). Pirkei Avos includes also the Luchos and the stylus used to write them (Rashi; Rambam, R. Yonah
Why did the ass use the expression "Shalosh Regalim" rather than "Shalosh Pe'amim"?
Rashi: It was hinting to Bil'am that he was wasting his time going to curse a nation that celebrated the Shalosh Regalim (the three foot-festivals).