What are the connotations of 'Shirah'?
Targum Onkelos and Targum Yonasan: Shirah has connotations of thanks and praise (singing the praises of).
Why does the Torah write "Yashir" (in the future)?
Rashi #2 (citing the Mechilta): It refers to after Techi'as ha'Meisim. 4
Ramban: It is common for the Pasuk to speak in the future even when it is referring to the past. 5
Because the 'Yud' represents Machshavah (Rashi).
On other occasions, the person concerned had in mind to do something, and did not put his thoughts into practice - see for example, Melachim 1, 11:7 (Rashi citing Sanhedrin, 91b). The Ramban queries Rashi from a number of sources - See for example, Tehilim, 106:19 & 108:14 and from that entire chapter.
This is one of the places where Techi'as ha'Meisim is hinted in T'nach. (See opening Sugya in Cheilek).
Why does the Torah use the double Lashon "Va'yomru Leimor"?
Or ha'Chayim: Each one prompted the other to sing, until they all sang in unison. 1
Which is why "Ashirah la'Hashem" is in the singular (Or ha'Chayim).
What doe "Ki Ga'oh Ga'ah" mean?
Rashi #2: He did something that nobody else could possibly emulate - When someone fights against a man riding a horse, he can only defeat him by toppling him from the horse, whereas Hashem cast the horse together with its rider into the sea.
Rashi #3 and Ramban #1: However much one praises Hashem, there are always more praises to add - in stark contrast to a human king, whose praises are undeserved.
Ramban #2 (citing Targum Onkelos):
Seforno: Hashem is deserving of praise for the goodness that He possesses, and not Pharaoh, who claimed that the River Nile was his and that he made himself.
Rashbam: The term 'Ge'us' is often used in connection with victory in war - as it does in this case.
The horse is proud (vain) in battle, and the rider is proud over the horse (when he rides on his back), and Hashem was proud over both when He cast them together into the sea (Ramban).
Hashem does not tolerate people who are conceited before Him, and raises Himself above them and punishes them. Consequently, when Pharaoh, in his conceit, chased after Yisrael, He cast his horses together with their riders into the Yam-Suf and drowned them (Targum Yonasan).
Why does the Torah here write "Ramah ba'Yam" (with connotations of throwing up up) and in Pasuk 4, "Yarah ba'Yam" (which means He cast down)?
Rashi: This teaches us that the riders together with their horses were cast from the depths of the sea to the top and then down to the bottom again. 1