Irish home bases

I'm going to make a suggestion for the Antrim Coast.

Bayview Hotel in Port Ballintrae. One of my favorite hotels and its location is fabulous.
You take the train from Belfast to Coleraine, walk through the train station to the buses, and board a bus
https://www.causewaycoastway.com/travel/causeway-buses.php

It stops right beside the hotel. You will find yourself in a lovely town with a small harbor. It is walking distance to the Giant's Causeway and Dunluce Castle or you can take that same bus there as well as other locations along the Antrim Coast. I would do 2-3 nights here.

Hint: Spring for a view room. It is worth it.
https://bayviewhotelni.com/

Afterwards, we took a bus to Derry and then another bus to Galway to continue on with our trip.

Edited to ADD: You are trying to do far too much for two weeks on public transport. Narrow things down a bit. Dublin, Belfast, Antrim Coast, Galway, and one more and you will have spent your entire 2 weeks.

Yes, you will definitely need serious cuts to your wishlist to fully savor what you do (plus allow for serendipitous moments along the way). Even if you were driving, this list is more than ambitious. You must go with the promise that you will return to check more off your list.

My husband and I did a whirlwind week and a half in 2009 - Knowth/Newgrange to Donegal Peninsulas(2 or 3 days?)/Derry We took in the Ulster Folk Park and Museum on the way---Well worth it! We drove down to Dingle Peninsula for about 5 days, catching the Burren, the Cliffs and a few smaller museums off the beaten path. We did not get to the Blasket Islands. I think we had 2 or 3 overnight stops on the drive down from Donegal. We stumbled upon concerts in Dingle. (Could spend a few weeks on the peninsula) We had an afternoon in Cork before heading back up to Dublin. We spent no time in Dublin at all. We stayed near the airport on our way out, hit the ground running to Knowth on our trip in.

We returned in 2019 for a wedding, spending most of 3 weeks visiting sites we didn't get to before...still less time in each location than we wanted, but we managed Rock of Casel, Kilkenny (loved it), a couple of nights winding back up to 2 days in Dublin for wedding, Belfast (Thanks to Rick's recommends---take a taxi tour---we were very pleased with the "Catholic" driver and wished we'd had time for the "Protestant" driver as well.) Bangor, Antrim Coast, all across to a Donegal penisula, again. Down to Connemara and Galwaybefore back to a day or two going in to Dublin. (Westport is on our list for our next trip).

Oh, we did get Hill of Tara, and Newgrange (only Knowth was available on our previous trip---now it's wrapped up in one tour.)

Between Ireland's trips and a trip to England in 2018, my husband has logged over 4,000 miles on the other side of the road.

I think we managed several other items on your list, as well.

Definitely go to Westport. I did the bicycle thing that Rick's guidebook talks about. Fantastic ride. They drive you out to this island and you bike back to town. It was the highlight of my 4 week tour of Ireland. Even if you don't do that, Westport is a lovely town with great pubs. If you go to Galway, take a boat to the Arran Islands and stay over one night.