1. Yes it's like a browser game
2. Technically you only befriend pokemon in the dream world, then you can send them to your game and catch them there.
3. Not 100% sure but I think the pokemon you put to sleep gains some exp while in the dream world. (the same as pokemon gained exp in the pokewatcher)
4. Yes you can have friends and visit their homes. I think you can water their berry plots too...