Building a "Photo Booth" without the booth?
It's entirely possible in fact I built a commercial photobooth using similar parts :)   Check out gphoto for an open source remote capture solution ( that seems to be the industry term for click camera button over usb).   My experience was mostly with canon cameras and they were relatively easy to use.   I used a G10, and one of their rebel dslr's.  Canon also has an official sdk here

With either one you should be able to take a picture with flash and auto focus.   So then you just have to decide how to hook your button up.  Maybe something simple like this USB connected one?   Or if you're using PI you can just wire one in.  

After that just some software that uploads it where you want it to go should be easy.