2. I would use 1-1/2".
3. In the subpanel in the outbuilding you will have a terminal bar that is used for the neutrals and a separate terminal bar that is used for the ground wires. This is different than what you probably have in your house panel where the neutral and grounds share space on the terminal bar in your circuit breaker panel. The sub panel neutral terminal bar will be connected to the neutral wire that is running back to the house panel. The sub panel ground wire terminal bar will have a bare copper wire that runs to your ground rod. I do not believe you want a ground wire between the house and outbuilding since you have the ground rod, but I am sure someone will confirm or correct this.