When you enter Ireland, you will also get approval to enter the UK. How Irish immigration is handling the ETA is something you can report back about.
No immigration from Ireland to the UK but there is immigration, for Americans and other non-British or Irish subjects going from the UK to Ireland.
The Central Bus station is closer to T2 than T3. (I was there yesterday and many times before that.) As you exit the terminal, you will see numerous banks of elevators (lifts) some with big markings saying "Central Bus Station." That's where you want to go. It's a five minute walk if that.