Who informed Par'oh that Yisrael had fled?
Rashi, Ramban #1 (citing the Mechilta) and Targum Yonasan: Spies whom Par'oh had sent to spy on Yisrael. 1 When three days passed and it was clear that Yisrael were making no effort to return to Egypt, as they had agreed to do, they reported to Par'oh (on the fourth day) that they had fled. On the fifth and sixth days 2 Par'oh gathered an army and chased after them (See Sifsei Chachamim) and on the seventh day 3 he caught up with them by the Yam-Suf.
What led Par'oh to believe that Yisrael did not intend to return to Egypt - particularly bearing in mind that had turned back and were traveling towards Egypt?
Rashi, Ramban #1 and Seforno: Refer to 14:5:1:1.
Ramban #2: When Par'oh was informed that Yisrael had back-tracked and were now encamped in front of the Yam-Suf, he understood that they wandering aimlessly in the desert without making the least effort to find a fitting location where to sacrifice to Hashem - their initial excuse to leave Egypt. Moreover, they left Egypt with an air of triumph, 1 with joy and music, like slaves who had just been set free, not at all like slaves who had been given a short respite and who were then destined to return.
Oznayim la'Torah (citing the Mechilta): Itwas only after the spies returned and reported to Par'oh that Hashem ordered them to turn back - so that it should seem as if they were afraid to confront the Egyptians when they caught up with them.
As Pasuk 8 indicates (Ramban).
What caused Par'oh and his servants to change their minds?
Rashi: The 'money' that they lent them.
Seforno: They believed that had they consulted Ba'al Tz'fon, he would have assisted them to stand up to Hashem and they would have needed to allow Yisrael to leave. 1
Targum Onkelos nd Targum Yonasan: They were sorry for hsaving sent away Yisrael from serving them, as the Torah specifically writes. 2
Oznayim la'Torah: In fact, they deliberatelly avoided any mention of the money that they had lent Yisrael - a proof for Gevihah ben Pesisah, who claimed that the money that they 'borrowed' fell far short of what the Egyptians actually owed them for their four hundred years work.
Moshe asked to leave for three days. Why were Yisrael not returning like he said?
Riva (13:17, citing R. Yitzchak): He never said that we would return. He said, we will go three days - and more, if needed.
Hadar Zekenim: Par'oh gave permission to leave permanently; he freed Yisrael (refer to 12:31:1:3****).
Why did Egypt regret having sent Yisrael? They were forced to, due to the severity of the Makos!
Moshav Zekenim: They regretted that they did not stipulate that Bnei Yisrael may leave, but they will serve Egypt wherever they go. 1 This is why it says "me'Avdenu."
The verse says "va'Yehafech Levav Par'oh va'Avadav" - Hashem hardened their hearts to act insanely (based on\ Ramban, verse 4).
Why did they think that Yisrael would consent? Moshe did not consent that we leave even one animal, and even demanded that Par'oh give to us animals! (PF)
QUESTIONS ON RASHI
Rashi writes that after three days, Par'oh's spies returned and told him on the fourth day. How could they travel in one day, what Bnei Yisrael traveled in three days?
Riva (13:17) #1, Hadar Zekenim: On day three, Yisrael headed back towards Egypt, so at the end of the day they were only one day away.
Riva (13:17) #2, Moshav Zekenim: We had said that we will go three days in the Midbar, which is 120 Mil. The spies saw that we went this far on the first day, from Ra'amses to Sukos, and we did not start returning on day two. 1 On day two the spies started returning, and reached Par'oh at the end of day four.
Riva (13:17) #3: Individuals can go faster than big caravans. 2
Why did they start returning then? Perhaps Yisrael intended to return on day three, the same distance that they went on day one! How did they know how far they went through a miracle (Kefitzas ha'Derech, or on eagles' wings - refer to 4:5:152:1)? Perhaps they knew the distance between Ra'amses and Sukos. (PF)
Rashi writes that Par'oh's spies told him that Bnei Yisrael were not returning. How could spies keep up with Yisrael? Above (12:37), Rashi said that Yisrael went from Ra'amses to Sukos (120 Mil, three days journey) in a flash!
Riva (13:17, citing R. Meir Kochvi): Hashem carried also the spies on eagles' wings, just like He carried the Erev Rav, to show them the miracle.
Rashi writes that on day seven we entered the sea, and we sang Shirah in the morning. In Bamidbar (15:41), he wrote that the eight threads of Tzitzis correspond to the eight days from leaving Egypt until Shiras Yam Suf!
Hadar Zekenim: Here, Rashi counts from when they left Ra'amses. In Bamidbar, he counts from Erev Pesach, when a. Yisrael left Goshen and gathered in Ra'amses, and b. they prepared to leave, through Shechitas 1 Korban Pesach. They knew that they will leave the next day. Da'as Zekenim (16) - they did not make all the preparations - "v'Gam Tzedah Lo Asu Lahem."
Moshav Zekenim citing R. Menachem of Yu'ani: The eight threads correspond to the eight days that Yisrael delayed at Yam Suf to take booty. 2
Mizrachi (Bamidbar 15:41): Rashi there cites R. Moshe ha'Darshan but Rashi himself disagrees.
Riva (13:17): A support for this is Devarim 16:6, "Tizbach Es ha'Pesach... Mo'ed Tzeischa mi'Mitzrayim." He considers these two separate answers. Da'as Zekenim (16) and Sifsei Chachamim (Bamidbar 15:41) bring only the latter answer. Moshav Zekenim (Bamidbar 15:41) says that the redemption began with Shechitas ha'Pesach. Here he says that he counts from the day of eating Pesach. (I.e. if we say that day begins in the morning, they ate on the 14th. - PF).
Via a small change in the text, one can resolve this with Rashi in Bamidbar (PF).