Pls help with puzzle for itinerary 8-9 days Ireland

Neither of you can fly direct to Belfast, Neither Belfast International or Belfast City airport has any long haul routes, and neither has any short haul routes from Germany.

The train from Dublin to Belfast is hourly and there are 3 bus companies provide hourly service from DUB to Belfast.

I think I would fly in to DUB then immediately transfer to Galway by direct bus or train (from Dublin Heuston, direct buses go there from the Airport).
Then Day 2 Cliffs of Moher take the Bus #350 to Moher at 0800 or 1000 arriving around 2 hours later, returning at 1121, 1321 or 1521 from Moher then train or bus late afternoon to Dublin.
Then a few days in Dublin
Transfer day morning train to Belfast. Leave early enough and you can now be in Belfast by 0800.
After time in Belfast bus back direct to Dublin airport- at least the Translink bus runs 24/7. If the flight is early get a late pm bus down, and book an airport hotel for the night.

You will also need a UK ETA for your trip to Belfast- https://www.gov.uk/guidance/apply-for-an-electronic-travel-authorisation-eta
even though no one will check it, crossing the open border

I understand not wanting to do a group tour. But a day tour to the Cliffs might be the easy answer. There appear to be several options from Dublin. If money is no object, I’m sure a private tour would be possible.

If you’re open to possibly staying closer to the Cliffs of Moher at all, you’d get an opportunity to hear music (County Clare is arguably the best place in Ireland for that), have great food, and an opportunity for further exploring, including a small village.

Galway is my favorite city in Ireland, reachable by comfortable bus (coach) from the Dublin airport, or downtown. Even closer to the Cliffs is the charming village of Lisdoonvarna, just down the hill from them. We stayed at a small B&B just outside of the village 15 years ago, and actually rode bicycles steeply from there up to the Cliffs of Moher visitor center, but a cab from Lisdoonvarna would be easy to get. We ate dinners twice at the Wild Honey Inn in the village, which was known for its outstanding food then, and is more renowned now, having earned a Michelin Star, plus being a place to stay. There’s a Burren Smokehouse in town with superior smoked fish to sample or purchase. The Burren National Park to the east has unique terrain and plant life, and could be explored with a naturalist guide, who might provide transportation to get there.

Down the road from Lisdoonvarna is tiny Doolin, long known for its music pubs, but Galway city wouldn’t disappoint, either.

Just my two cents, but if I was only there for eight days, and the whole visit was going to be distorted for the sake of seeing Moher, I'd reconsider. There are lots of gorgeous sea cliffs, and the others have the advantage of not having to wait in lines and jockey for a view. We finally gave in and visited Moher on our fourth trip to Ireland, and we only did it by boat. And that was only because we were in Doolin to watch Blackie play the pipes (on a Wednesday) at McDermott's. I'd also reconsider renting a car. I'd much rather drive in Ireland than NYC, and driving on the wrong side becomes natural by the time you get out of the airport, although I wouldn't head straight into Dublin city-center without a few hours to adjust - in fact I try to avoid driving in Dublin altogether.

If by musicians, you mean you're interested in listening to or participating in trad sessions, then I might consider visiting Doolin, which lends itself to including Moher. Participating in sessions probably means getting out of Dublin/Belfast because most music is organized groups playing popular drinking songs and ballads, although there are less formal settings if you look hard enough (e.g. The Cobblestone). If you get out west to Donegal, Sligo, Doolin, Dingle etc. you'll find yourself being invited to join if you carry an instrument.

Our trips are based on finding good trad as many nights as possible. Last trip, I only missed on two of nineteen nights (because I trusted Facebook). We also went to Temple Bar for the first time, and were very disappointed. Luckily we found good stuff on the walk back to the B&B.

I flew Lufthansa from Frankfurt to Belfast City this summer, so there are flights. That said, I chose the flight for the scheduling. It was more expensive than flying into Dublin.

Bean, I’ll toss out a couple of thoughts. Consider landing in Dublin and staying 1st night. Quick train to Galway the following morning. Great small city, good pubs usually able to find some traditional music being played. Stay in Galway. Next morning a bus to Doolin, boat to Cliffs and stay over in Doolin to enjoy some great music. Head back to Galway, stopping to explore the Burren area then continuing on to overnight in Galway. Train back to Dublin and split the time between Dublin and Belfast, both great cities.
It’s a bit of moving around but it checks most of your boxes.
Dublin
Belfast
Cliffs
Small Village
Exploring and Music.

The cliffs are cool, but seriously, there's other stunning cliffs. We actually preferred the Kerry Cliffs and there weren't a bazillion people to deal with!
Definitely would try to hit some smaller places...I can't wait to get back to Doolin and Galway!

The Cobblestone pub in Dublin is THE place to hear (and join?) trad music. We got there and back by taxi, and it was a great experience.

Temple Bar in Dublin was not just disappointing, but disgusting. Walking at night, there were wet streaks on most sidewalks, yet it hadn’t rained. It dawned on me that it wasn’t water, and there were drinkers who couldn’t be bothered to find a restroom indoors before relieving themselves outside.

You state you are musicians in passing. Was not a whole lot of emphasis there. Or is it very relevant to you? In which case a visit to Galway would give you an opportunity to try different pubs and music very easily. The town is lined with them playing all through the week.