A friend and I played a game recently. He only had the starter set, and I had the starter and the Reliant. We both ended up using Picard on the Enterprise D in our build. How do we determine which Picard gets to go first since both have the same skill rating and both are federation captains?

Rulebook, pg. 18 Initiative, last paragraph. Basically, you'll do a die roll at the beginning of the game, whoever wins it will be the one who has initiative over the other for the rest of the game, in your situation.

Depends on what rules you are using.

If you're using the squad building rules in the advanced rules, you first check to see who's fleet is the smallest (ie: a player with 99 would have INT over a player with 100). In the event of a tie then you check for faction followed by a roll-off as required, as DangerousFat indicated.