Not the best itinerary and no, public transport in Ireland especially if you want to see the stunning west Coast is NOT the way to travel.
So 4 full nights in London, train to York, 2 nights.
Drop Edinburgh.
Train to airport Manchester then fly from Manchester to Dublin or Belfast ( whichever is Cheaper). If flying into Belfast stay 2 nights at any of the 3 Premiere Inns. Read TripAdvisor reviews to see which appeals to you most.
See Titanic exhibition, visit St Georges Market, City Hall, have seafood chowder at Mournes, tour Crumlin Road Goal and schedule The Troubles Black Cab tour. Pick up your pre reserved rental car at the airport and begin your drive West.
If flying into Dublin stay 2 nights and wander the city center, visit Book of Kells, people watch in St Stephens green, visit the national Museum of Ireland, and Christ Church Cathedral. Pick up rental car at Dublin airport and off you go.
Powerscourt, Glendalough, Rock Of Cashel
Overnight in Kinsale. Day 1.
Next day head off to Dingle ( base in Dingle). Can do ring of Kerry, Gap of Dunloe, Slead Head drive.
It’s 4.5 hours back to Belfast or Dublin. If you rented car in Belfast NI drop of rental then its an hour bus or train ride back to Dublin for you flight home.
Driving on the opposite side of the road is not that daunting. Takes but a few minutes to adjust.
I was born when Truman was in office so driving with a clutch no big deal. Have driven solo throughout Eire and NI twice. Simply get the smallest most comfortable car and All the insurance you can. Plan your routes the night before ( either old fashioned map or online via Google satellite view making note of town names so you know you are heading the right way), and remember you will need British pounds as currency in NI and Euros for Ireland. Use bank ATMs.
Then one thing that seems to confuse American drivers are the roundabouts. That’s why it’s smart to copy down the names of towns. Helps you know which exit to take and if per chance you miss it simply go round again and take it.
Enjoy your travels.