Being the 'great outdoors' - yes, there can be mosquitoes, but to be honest, in all our years there we have never been bitten. Maybe we just don't taste so good! I have heard others say they have been bitten, so I would be prepared just in case they find you 'tasty'!!!
I'm sure you can request a play pen, but not sure about an extra bed. I would call and ask on that one!
As for a rec center, there is a game room (smallish), but to my knowledge nothing on a large scale.
From what i've heard there is really no room for an extra bed, but several have reported putting a pack n play (available) in the closet recess. there's a sofa and nurphy bed in the living area, and a double and set of bunks in the bedroom.
1. Most people report that mosquitos are not a problem. I think WDW does a very good job managing the situation.
2. There is a pack and play in the cabin but I don't think they have extra roll around beds. The cabin already sleeps 6 and thats the max unless you have someone under 3. It does have a small couch.
Most would say the entire resort is the best rec center around.