Europe in August

Your list of places is way to long to see in two weeks or 10 days. Eliminate about two thirds of the places and enjoy the places you visit instead of spending a lot of time in airports or train stations.

In my opinion, you can go to the beach in the USA or Caribbean a lot cheaper than flying to Europe to do that and miss all the history, art and culture.

Not sure why you didn't book round trip tickets, but if you fly back to the USA from Britain, the taxes on the fare are very high.

Use Kayak to check fares.

You will have to pick a direction and go that way. Even if you choose a couple of destinations, the travel between the two eats up your time and money.

The Atlantic Coast of Northern Spain, Basque territory, is nice in August. Most of the rest of the year it's rainy. Check out San Sebastian/Donostia. You could take day trips inland from there to Pamplona (and a castle I love in nearby Olite). You can also get to the heart of the Rioja wine region as a day trip or short excursion. Go up the coast into Basque France or the other way to see Bilbao (great museum) and Santander (more beaches).

Stockholm is nice in August, but I much prefer Copenhagen. IMO there is much more to see and do. I visited a family beach on the north coast of Zealand in late July. It was nice and the water was much warmer than Cinque Terra in October. You can also take a short Scandinavian cruise or just ferry between Scandinavian cities to get out on the water. July/August (maybe late June) are the only times I'd visit Scandinavia because the days are long and it's the most reliably nice weather.

I avoid most places in August because of the hot and crowded cities.

Even before seeing Brad's note, my first instinct was to suggest Denmark. In a very compact country, it has all the features you've mentioned and probably more moderate weather. And the benefit of not being any one's "first choice," so nobody wins or loses the discussion. Family rooms in hostels can be a good budget option and some are right on the beach. It's an easy country to drive around, if driving in Ireland was worrying you. But of course, Ireland is similarly a small country that ticks all your boxes.

Yes I realize in two weeks that would be impossible.

The OP already knows all those destinations are too much, she is looking for options on pinning down a place in addition to Dublin that would satisfy both daughters.

Copenhagen would be a good option in August as noted by a couple of posters. One can take the train towards the airport and there are stops along that route with easy beach access. I didn't go just to lay in the sand, but they seemed like nice beaches. I can give you more details if you go that option. The other daughter can explore nearby Roskilde, its cathedral filled with tombs of almost Viking kings and a great Viking ship museum. Helsingor up north is a delightful town with a great castle. And many other options.

Barcelona would be another city that fits both daughter's interests. Easy, sandy beaches and lots of museums and sites to see. Plus nearby beach towns along the Costa Brava. Nearby Montserrat is a hikers dream that everyone might enjoy. But expect it to be very crowded in August. That would be the downside.

Croatia is a third option that would fit. Lots of historic sites in cities along the coast and the Adriatic coast is lovely. You can spend a week easily along the coast. Also expect crowds.

I'd pick one other destination from Ireland and split your time that way. Any of the above options offer plenty of daytrip opportunities to spend a week.

We are flying into Dublin because that was the cheapest airfare we found. 3 of us have been to Dublin/Ireland so it was just a stopping point. One scenario we are thinking is fly into Dublin, then on to London for 2 days. London to Barcelona and spend a week in Spain(Northern area only), then take a train to Belgium/Netherlands for 3 days. Good idea or suggestions on where to go after Spain. Husband and I have been to Germany and Eastern France (no one really wants to go to France). Husband and I have been to Copenhagen and had a 1 day layover in Stockholm. Thanks for the suggestions/help so far.

OP's older daughter (castles/museums) here. Thank you to those giving helpful feedback. As my mom said, we are very much aware we can't go to all or even half those places in 10 days. The hope was that those who have been could say yes, this location can make everyone decently happy without too much effort.
To update: we snagged a deal for $65 flights from Boston to Dublin, hence the return flexibility. Ireland is lower on the list as 3 of us have been there before and recently enough that we'd like to explore elsewhere this trip.
My sister has accepted she probably isn't getting the beach. Our dad is too fussy about heat.
In a way, we will be planning separate trips, just staying in the same place.
Mom just likes to walk all over. There doesn't have to be a destination so much as the ability to wander. But outdoor options are nice. Sis is similar to mom but likes nature even more. She does have the one year old. Dad's kind of horrible to travel with, but we're stuck with him. He's of the sort to complain about things being done differently there and tell everyone about his unique experiences at home. We don't plan for him, but we do plan around him. Lower crime areas and not too hot are his limitations. (Spain is too hot-hence no beach-Southern England is fine). I like history and a good bar. I can be just as happy at a small graveyard in ruins as at a massive museum. I'm probably the most flexible. I know this isn't my trip, I'll do what I want another time.

Now: the Denmark suggestions are positive and seem promising. Both parents have liked the idea of the Netherlands and dad/me want to see Belgium. Would those three be feasible for our setup and timeframe?
Is there an alternative to Copenhagen/Amsterdam/Brussels? For comparison, we're a family that loves the Galways of the world more than the Dublins. If those cities give that small town meandering feel anyway, awesome.
Thanks.