I haven't lived in a studio, but my boyfriend has what's called a studio in NYC (the kitchen is somewhat separate, but is open into the other room). I haven't noticed any problem. My first apartment had a separate kitchen, but there were no doors between rooms except the bedroom (which I kept open anyway), and it wasn't a problem either.I'm in a dorm room at the moment. It's good that the kitchen is separate so the room dosen't get stuffy. Have you lived in a place where the kitchen is in the same room as the bed and study desk? In other words, have you lived in a studio?
The studio is furnished and what's more it's actually a little 'house' at the back of a couple's house. They've got young kids just like you described, not too young as I saw a bike with training wheels. So no one to share the walls with. However, if the couple or kids go into the garden then noise can be heard.
If you're worried about it becoming "stuffy," two things to look for are the windows (opening a window will really help air the place out...even if sounds from outside bother you, you can leave it open a crack whenever you're out for a little while), and the vent fan...open the cabinet above the stove. If there is a duct there leading out somewhere, your vent fan is properly vented. If there is no duct in that cabinet, it's one of those weird, recycling fans that just filters stuff, but doesn't really vent it out. You want a kitchen fan that vents outside.
You're not going to find anyplace more silent than what you're describing though. Even with your own house, there are neighbors or kids playing in the yards, people's dogs barking, etc. If you find these noises very distracting when studying, you might want to just use earplugs to muffle the sound. At least you know you won't be disrupted from sleeping at night (I don't like wearing earplugs at night, since it's not comfortable for me to sleep and I worry about not hearing the alarm in the morning).