How does one translate "Azi ve'Zimras Kah, Vay'hi li li'Yeshu'ah"?
Rashi #1 (citing Targum Onkelos) and Targum Yonasan: My strength and my praise 1 is the fear of Hashem 2 (which extends over the whole world -Targum Yonasan); He issued the command and it became my salvation.
Rashi #2 and Seforno: The might and vengeance of Hashem was for me a salvation. 3
Ramban (citing Ibn Ezra): My might is Hashem and Hashem is my song (of praise), for He was my salvation. 4
Rashbam: Hashem is the might and the praise of Yisrael; He was for me a salvation.
See Rashi's objection (on grammatical grounds) to this explanation.
Because by casting the horse and rider into the sea He revealed His might, that He is King of the world, and that it is therefore befitting for those who were saved to praise Him with song, in their rejoicing to be servants of the King of the world (Seforno).
See Ramban, 'al Derech ha'Emes'.
Bearing in mind that throughout the Torah, Moshe uses Hashem's Name of 'Havayah', why here does he refer to Him as 'Kah'?
Ramban (citing the Midrash Tanchuma): Hashem swore that His Throne and His Name will not be complete until the seed of Amalek is wiped out. 1
See Rashi later (17:16).
What did they see that elicited the exclamation "Zeh Keili!"?
Rashi: In all His glory Hashem revealed Himself to Yisrael (at the Yam-Suf), and they pointed at Him with the finger. 1
Targum Yonasan: The babies left their mothers' breasts to point out the Shechinah to their fathers, and to tell them that this was the G-d who had fed them honey from the rock and oil from the hard rock. 2
Hence Chazal said that a maidservant at the Yam-Suf saw more than the prophets (Yechezkel ben Buzi) - with regard to the actual vision, but not as far as the depth of understanding is concerned (See Sifsei Chachamim).
With reference to when, in Egypt, their mothers had gone out to the fields, and, after giving birth to them and leaving them there, Hashem had sent an angel to bathe them and tend to their dressing (and fed them in Person).
And what is the meaning of "Ve'anveihu"?
Rashi #1 (citing Targum Onkelos), Ramban (citing Ibn Ezra) and Seforno: I will build Him a sanctuary. 1
Rashi #2: I will relate His beautiful characteristics to the whole world. 2
Targum Yonasan: "And we will praise Him.
Gemara #1 (Shabbos, 133b): The Torah is teaching us here that we should perform Mitzvos beautifully. 3
Gemara #2 (Shabbos, 133b): One should aspire to be like Hashem - to be gracious and merciful like Him. 4
What is the significance of "Elokei Avi"?
Rashi: My connection with Hashem did not begin with me; it is a legacy that I received from my fathers.
That He is awesome in His greatness (Midas ha'Rachamim) and in His Hashgachah (Midas ha'Din [Seforno]).
And what does "Va'aromamenhu" mean?