When you have a mixed fleet, how is initiative determined? I don't see it in the rule book and I havn't found it online.

For example, let's say I'm running Voyager with Donatra, Ch'tang with Picard, and Vo with Dukot for 100 points (I know, that's not right, but let's pretend). How is initiative determined in that fleet? Do I just get to pick? Would anything prevent me from just declaring the fleet as Dominion? I only ask for understanding, I play faction pure myself.

Faction is actually pretty much the last determining factor in initiative. You go by Captain Skill, so it would be:

Movement: Donatra, Dukat, Picard
Fire: Picard, Dukat, Donatra

I know the rule book seems to make a thing of factions on initiative, but it really only comes into play when you have equal Captain Skills.

What if two people are both running captains with the same skill and points? This happened to me: I was playing 100 point faction pure Federation including Picard, and the opposing player had 100 points including Picard on the Excelsior in a mixed fleet with two other Dominion ships. Who should get initiative? Or should we roll for it (which is what I think we did)?

From page 18:

"It may be possible for different players to have ships from the same faction with the same captain skill. When this occurs, determine randomly at the start of the game which of these ships will have initiative over the others throughout the game."

It doesn't specifically state you determine it by a dice roll, you could flip a coin, rock paper scissor, whatever. Dice roll seems easiest/most logical.

Just a side note, it's faction Of the SHIP the captain is a part of to determine order. So if Picard is on a Borg ship, and Picard on a Federation Ship, the Federation ship will move first as well as fire first.

(18 under Inititave order for reference)