1) Are they considered characters? Can cards that target characters target the monster?
To all intents and purposes, once recruited, treat the monster exactly as if it was a character. Cards that refer to monsters ignore it, and anything that can target a character or the party effect it. You can attach cards to monsters just as if it were a normal character card. When the monster dies, it is discarded and a replacement is not drawn from the character pool.
2) Do they count towards the 5 character party limit?
yes, regardless of where they come from. Any effect that would add a character or monster above 5, fizzles and is ignored.
3) Do you get to use their abilities listed on their card?
Somebody posted a question about a specific ability when a monster was in the party on the AEG boards. They couldn't remember the exact card, but one of the designers said if they could dig it up, they would try to give a ruling. Hardly official I know, but it does imply that monster abilities can be used in this way. I guess this will come down to common sense and how much time you want to give over to on the spot house rules. For the most part if it isn't obvious how it can be applied, it is probably best to ignore it.