Why did go Par'oh go each morning to the Nile?
Rashi: He made himself out to be a god who did not need to relieve himself; so he would go early in the morning (before anybody was around) to defacate (as he swam or paddled in the water
This is also the opinion of the Gemara in Mo'ed Katan, 18a
Why did Hashem pick specifically that time for Moshe to confront Par'oh?
Oznayim la'Torah: It was the ideal time to confront Par'oh
Why, for Makos Dam, Arov and Arbeh, did Hashem instruct Moshe to go and meet Par'oh when he made his way down to the Nile?
Ramban (on 8:15): For Makas Dam, because it was the first Makah, and Hashem wanted Moshe to confront him fearlessly as he made his way there in a show of strength. 1 And for Arov and Barad, which caused widespread death, he wanted him to warn Par'oh in front of the many people (who customarily would accompany the king on his stroll), to give them the opportunity to prevail upon Par'oh to submit to Moshe's demands. 2
Da'as Zekenim and Hadar Zekenim (26): These were new warnings which were not preceded by a Makah about which he was warned.
Zevachim, 112a #1 (citing Resh Lakish): He is Rasha; therefore go and greet him with Chutzpah. 3
Zevachim, 112a #2 (citing R. Yochanan): He is a king; therefore go and greet him civilly..
According to those who maintain that Aharon's staff was not synonymous with that of Moshe, why did Hashem instruct Aharon to take Moshe's staff which turned into a snake in front of B'nei Yisrael, and not his own
Oznayim la'Torah: Because it was on Moshe's staff that was engraved "D'tzach, Adash, Be'achav' 1
Refer to 7:26:1:2 and note 2. The Oznayim la'Torah adds that by eachy of the three groups, the first warning took place when Par'oh went down to the Nile ("Lechy el Par'oh") and the second Moshe went to him in his palace ("Bo el Par'oh"). See Oznayim la'Torah on Pasuk 17 DH 'Hinei Anochi Makeh'.