Question 1: I had this come up, my hero completed his movement and turned over the flooded area token. The placement of this token was critical to the game because it was on a part of the map that was 1 tile thick, so it was the only way to get across the map. The rules state that flying units may enter this area anytime. I was wondering if they control the area when they move in and if so can I build a stronghold there so I can recruit more flying units. We had also wanted to know if we could place a rune here.
All the flooded area token does for an area is make all 6 borders water borders. It doesn't affect the area (or the units in the area) in any other way. Thus, yes, you can build a stronghold there. Heck, you can even recruit non-flying units at the stronghold (though they won't be able to leave until winter, or unless a Cryomancy token appears). And yes, you can put a dragon rune there.
Question 2: The frost tower quest card.
I did this in a free city, one that produces a rune if you own it. This card says to spend 6 influence and then gain a dragon rune, when this happens do I place it in the same tile as the city? I ask this because normally you can not have more than one rune in any one area.
Since it uses the phrase "gain a dragon rune", the regular rules for that apply: You can remove up to 2 fake runes from your areas, and then you place a real and fake rune token in any friendly or empty areas that don't already have rune tokens. You are not required to place either of those tokens in the city area that the quest was completed in.
If the quest meant for you to put a rune in that area specifically, it would use the phrasing "Place a dragon rune in this area."
(IE, "gain a dragon rune" and "place a dragon rune" are different)