[Pharoah] called to Moshe and Aharon at night and said to them, "Go depart from my people, you and the descendants of Israel, and go worship Hashem as you spoke." (12:31)
It was told to the king of Egypt that the people [of Israel] had fled, and the hearts of Pharoah and his servants changed in regards to the people, and they said: "What have we done, for we have discharged them from our service!" (14:5)
Why would the Egyptians be angry at Israel for fleeing, when they themselves told Israel to leave in the first place?
Contrary to popular impression, Moshe never asked Pharoah to leave Egypt permanently. Though many of the requests were ambiguous (and Pharoah undoubtedly suspected they wanted total freedom), Moshe technically only requested to leave Egypt for three days to "worship God in the desert". (See R' Yisrael Rozen's recent article in Shabbat Beshabbato.) When Egypt told them to leave, they did not mean to leave permanently, but only for as long as was needed to appease God and stop the plagues. Thus the Egyptians felt tricked upon realizing that the Israelites had not stopped at a three-day-journey distance. Instead the Israelites kept going, and this was the "fleeing" which Egypt was angry about.
This may be the reason why the splitting of the sea is traditionally held to have happened on the 7th day of Pesach. Israel left Egypt on the first night of Pesach. If they had traveled three days into the desert, spent some time sacrificing animals, and returned - as the Egyptians expected - they probably would have arrived back in Egypt on the 6th or 7th day of Pesach. Only at this point, when no Israelites showed up, did the Egyptians realize that they had been deceived. They then sent chariots to attack Israel (chariots travel much faster than Israelite civilians), and the rest is history.
UPDATE: Verse 14:3 shows that the chronology is more complicated than what I just presented. Nevertheless, the 7th day is still a reasonable guess as to when the splitting of the sea took place. And a reasonably guess is all I'm looking for - obviously there won't be any proof.
The special tune for Az Yashir
Which lines of Az Yashir get sung in the special tune and which don't?
I'm convinced the criterion is the presence of God's name. If God's name is part of a verse, that verse gets the special tune. If God's name is near the beginning of a verse, the last few words of the previous verse get the tune too.
We may gain further insight by examining which name of God gets the special tune. God is referred to by a variety of names in the Torah. At first glance this seems redundant and confusing. But God's names fundamentally do not refer to God as an entity, but rather to certain patterns of behavior manifested by God. A good example is the expression "know my name", which does not mean "remember what's on my nametag", but rather "recognize me and my power". When God says, "The Egyptians will know that I am Hashem, when I honor myself by [destroying] Pharoah and his chariots" (14:18), it does not mean that the Egyptians get a vocabulary lesson. It means they get to experience the miraculously destructive "personality" which the name Hashem represents.
There are two "basic" names for God in the Torah - Elokim and Hashem - with the other names mostly derivatives of these two. Generally, the name Elokim generally represents God as manifest through the natural workings of the world, while Hashem represents God's miraculous or particularistic interference with the natural order. It is easy to see (and this is stated explicitly several times, including verse 14:18 quoted above) that Yetziat Mitzraim is the quintessential example of the "Hashem" aspect of God's activity. Within Yetziat Mitzraim, the splitting of the sea is the best example of all.
Not surprisingly, then, the name of God used in Az Yashir is almost exclusively Hashem. By singing all the verses which include Hashem (and almost no other verses), we are not only accepting God and offering praise for the miracle, but also specifically acknowledging the "Hashem" aspect of God's activity, which is clearer here than at any other point in Jewish history. The point is to demonstrate that we, no less than the Egyptians, have "learned" Hashem's "name".
And Amalek came...
For the first time ever I will post a link to a relevant post I made in the past. I think it is possibly my most insightful post so far, though perhaps not the best stylistically. (By the way, the last sentence of that post was misleading and wrong until I recently revised it.)