1. It's your choice, you can change the address for each transaction, or always use to same (to receive coins of course, when sending the client automagically creates some new addresses anyway). No, others can't see all your addresses.
2. Having 2 different wallets doesn't solve your problem.
If I know that you own A, I can see that you moved all balance from A to B, so I can guess that B also belongs to you. Then I only track B instead of A and you are right where you started.

To stay truely anonymous only connect to the network thru TOR, change the address for each transaction and use some mixing-service.
Combining all that, split and merge transactions a lot and you should be fine.