10 days in Ireland

Looks like a good trip, though it's hard to tell how long you plan to spend in each place. If you don't plan to start in Dublin, then you might consider Aircoach bus service direct from DUB airport to places like Cork, Galway, or Belfast, saving you a ride into Dublin to get the train to somewhere else. Dingle is a great place for a couple of nights, especially if you can get a tour or driver to take you around the Slea Head route, or a boat out to Great Blasket Island. Galway is a good base for the Cliffs of Moher and/or Aran Islands and/or Connemara, which would mean spending two or more nights there. Galway itself is interesting and fun but its main value, I think, is as a base for other places.

I did a RS tour last year and enjoyed it all, including Belfast. I was glad to be exposed to "both Irelands," whose interlocked history is so interesting, but if I could choose just one it would be the republic. I'm not sure you can get train service from Belfast "down and over the west coast and southern portion by Cork" (i.e. counterclockwise around the island), see what Irish Rail offers. Generally the trains run between Dublin and other places, not around the whole island.

Also consider flying between Cork and Edinburgh, or Belfast and Edinburgh, or either and Glasgow, instead of returning to Dublin just to get to Scotland. And you might consider "multi-city" flights from your home to Dublin and back from Scotland to home, rather than a round trip between home and Dublin. That shouldn't cost much more, if any more, and could save you valuable time -- even if your return trip routes you through Dublin.

I can't help much on most of your questions, but regarding Belfast, I 100% it's worth the time. We were only in Ireland for 8 days and spent 2 of them in Belfast and we're happy we did. Specifically, the Black Cab tour, St. George's Market, and the Titanic Experience were all very memorable. As a plus, our best breakfast of the trip was at Harlem Cafe in Belfast.

Enjoy!

PQ

We just spent a few nights in Ireland & Northern Ireland in mid-September. Our trip was more of a reconnaissance. We were in search of family history. We rented a car for the 4 nights we were there. Here are my thoughts, I hope you find them useful. We landed in Dublin and picked up our rental car. Dublin can be challenging to get around by car and parking at times could be challenging also. IF you rent a car, and IF you park on the street, make sure to look for and obey the parking signs. Some street spaces you pay for, some give you a few free hours. Question 1: we had planned on those cities but they didn’t make the cut, so I can’t help you there.
Question2: Have you driven on the wrong side of the road before? (Readers, please withhold your right or wrong comments). Have you driven on narrow roads while driving from the opposite side of your normal driving position? I have done so before and I love the challenge. My wife, on the other hand, could not get used to objects passing by on her left peripheral vision. IF you rent a car in Ireland and want to drop it off in Northern Ireland, try to save yourself the drop fee and get a rental with UK plates. If you do that, be aware that the speed limit is in kilometers per hour in Ireland, and miles per hour in Northern Ireland. The little country road we crossed the border on was only marked by a sign with a black circle.
Question 3: Yes to spending more time in Belfast. I highly recommend watching the 5 part documentary shown on PBS “Once Upon A Time In Northern Ireland”. We were only there for one night. We did the hop-on hop-off bus tour and then the Titanic dock tour before we headed to Belfast city airport to catch a flight to Amsterdam to get us home. I also recommend that you rethink your flight plans. Good luck with your travel plans.

We loved Belfast and are going back on our next trip.

Public transportation "vertically" is very clumsy, as the trains are all "horizontal".

We've always rented a care in the Isles and have never had nightmares. It takes a few moments to adjust to the driving on the left, and a few more intersection experiences to figure that out, then it will be as natural as driving at home. Having a car opens up so many destinations.

With only ten days, encompassing Dingle and Belfast in the same trip is probably an over-reach. I'd focus on one or two regions with that amount of time.

We just spent 2 weeks in Ireland without a car. We chose to stay in the southwest rather than try to get all the way north within that time frame because of the "hub and spoke" nature of the train routes. We did not spend time in Kilkenny because we had some other specific things we wanted to see but would definitely spend time there next trip. We spent 3 nights in Dingle and took a great van tour of Slea Head. We flew into Dublin, out of Cork to the UK and back into and out of Dublin, all Aer Lingus flights. If you haven't spent time in Dublin before, I would definitely recommend a few nights there. I'm not a big city person and I loved it. We have taken the Northern Ireland/Scotland ferry (in to opposite direction) but it was 50 years ago so no input there, lol. Hope there's something helpful in this.

If you are not going to have a car then I would not try to get to Dingle on public transit. It will eat up far too much of your time, and then trying to get from there to Galway will be even more time consuming. If you want to see that area then use Killarney as a base as it is easy to get to by train, and there are many guided day trips you can get from there. Still, even if you use Killarney as a base it will take you half a day to get there by train, and you don’t have a lot of days on the ground. From Dublin the two closest options for home bases are Galway and Belfast. I spent four nights in both, and took several guided day trips to see the sights and was never bored. You can do Kilkenny as a day trip from Dublin by train, or spend the night there and then take the train to either Galway or Belfast. No matter where you go in Ireland you will wish you had more time so try to limit the amount of time you spend on a train or bus as you don’t want to see the country looking out a window. Fly home from Edinburgh, or Glasgow, not Dublin.

Don't try and cram too much into your ten days. Have you made allowance for travelling between places? Kilkenny to Dingle is going to take a full day by public transport. Galway to Belfast is going to take most of the day, especially as you are likely to have to go via Dublin. Dingle will take even longer...

Another option to be aware of is the availability of cheap, direct, flights between Dublin and the Kerry airport near Killarney (and Dingle) on RyanAir. There are two per day in each direction: departing Dublin around 9:40 AM and at 4:10 PM. The return flights back to Dublin are in the AM and PM too, and the flight itself takes less than an hour.
If your arriving flight into Dublin would allow, you could hop on the morning flight to Kerry and then base yourselves in either Dingle or Killarney for a few days to avail yourselves of the numerous day tours available from one or the other, then return to Dublin to finish out your time there, as well as being central for easy day trips by bus or train to Belfast, Galway, or Kilkenny..
Not sure how you've formed the opinion that driving in Ireland is nightmarish but, quite frankly, it isn't. While having a car in any of the cities would be a royal pain, driving in the rural countryside has always been a pleasure for us and I can recommend it without hesitation.
There are rental car outlets at the Kerry airport if you decide to change your mind.