I know the rule book indicates picking one of two randomly chosen characters, but just to be clear...
1. Are the character sets intended to be specific to each expansion?
2. If we're using all 61(?) characters, do we still only choose two?
The scenario I'm specifically thinking of is if we're playing any mode that has characters that are tailored to that specific mode, the probably of getting a character that is useful for that mode drops substantially. Is there any ruling on this?
There isn't. That is just a recommendation for opening the set for the first time. They only give players 10 new characters per set and given 4 players that is the only way you can fairly deal them out. After you have obtained two or more sets its not uncommon to deal out 5 characters or even draft 5 characters each.
Pretty much depends on you.
What I do is to take all character mode specific from the character Pile to choose From. Then shuffle that pile of character cards and draw 10 cards without looking and set them face down on the table. Then I allow players to choose 2 of them and then proceed with the game. If playing partner modification mode, what I do is take out the facedown cards from the game and take new 10 cards from the character pile and do the same process to choose the characters. When playing a game like outbreak or Partner modification mode I add the characters whose effect is specific to that mode and then get the cards from the character pile until I have 10 again on the table (Of course then shuffle it to avoid people knowing which character is which). Thats pretty much how I do it.