Well, you don't have to go through Dublin City to get to the European Club. There is a motorway, the M50, that brings you around the city to Wicklow and the southeast. If you are only playing links courses while in the east/southeast, then your options are Baltray, Seapoint, Laytown & Bettystown, Portmarnock, Portmarnock Hotel and Golf Links (the Bernhard Langer one), Corbalis (a public links course €12 for some great holes, also right beside the Island), The Island, Royal Dublin, St Anne's, European Club and Arklow Golf Club.
Laytown and Bettystown has improved its layout and is much better now. Seapoint is a new course right next door to Baltray that has some excellent finishing holes. Baltray, European, Island and Portmarnock are brilliant. Indeed, Portmarnock is one of the cleverest courses you'll ever play. I have never played the Bernhard Langer course at Portmarnock, but it is supposed to be good. If you are traveling to the European Club, you could also visit Arklow, which is a gem. A very old-fashioned layout and very entertaining.
It's a pity you are not visiting the west coast, cause there are some brilliant links courses there, like none you'll play anywhre else.
Anyway, enjoy your trip.
Bushmills is a very intimedating place if you are from the south of Ireland, with all the Union Jacks. but I'm sure it would be very interesting place to stay. It's home of the oldest licenced distillery in the world and produces some of the nicest whiskey to be found anywhere.
If you are planing to go out for a few pints and some music in Dublin, I would recommend Baggot Street, very close to the centre of the city. Lots of great pubs, especially Donneny and Nesbit. Then there is Johnny Fox's in the Dublin mountains, which is great for music and seafood and very popular with visitors.