Ireland and the UK have a Common Travel Area agreement, so most times, traveling from Ireland to the UK, or UK to Ireland, there is no immigration or passport checks (other than as ID).
So yes, you arrive in Ireland from the US, you will go through Irish Immigration in Dublin (passport checked, maybe stamped if they still do that).
Dublin to London flight, you go through security, and have to show your passport, but no separate exit immigration from Ireland, On arrival in London, you should just be able to exit the airport without immigration. Should qualify that as "nearly always", some airports and some airlines may implement immigration measures beyond the minimum.
As for time to allow, you could get by with the two hours typically used for domestic flights, as opposed to the three hours typically suggested for International flights. Of course, take into account your own comfort level, strange airport, how busy security might be, whether you have to check in vs just head to security, etc.