If you have "two weeks including travel" then you really only have 11 or 12 days. It is certainly not too long to visit Ireland + N.I..
Ensure you have hotel reservations in advance. Festivals, sporting events, and European (not just Irish) holidays can fill all the hotels, especially on weekends.
Fly to Dublin as you already plan to do. There are airport buses (6 Euros per person) and taxis. Warning: If you buy airport bus tickets in advance and arrive early in the morning then you must print the vouchers at home. The airport bus office at the airport may not yet be open when you are ready to go.
I have stayed at the Academy Plaza Hotel on Findlater Street (off O'Connell Street) several times. It is in a good location and both the airport bus and the hop-on/hop-off buses stop in front of the hotel.
You do not need a rental car while you are visiting Dublin. Exploring the city center on foot is practical. Use the hop-on/hop-off bus, public transit, and taxis to get around when you don't want to walk.
The hop-on/hop-off bus is great. I recommend choosing one of the buses with a live guide (instead of a recording) and simply ride the bus for the whole circuit the first time. It takes about 90 minutes and gives you a really good overview of Dublin. Then decide what attractions you want to visit.
On our visits we did light activities and mostly relaxed the day we arrived. Then alternated two days in Dublin wit day trips away from Dublin.
The Jameson Distillery (museum) has a good bar & restaurant you can get to without buying tickets to the museum. It is not in the city center but is on the hop-on/hop-off route and is a good place to stop for drinks/food. They do a GREAT hot whiskey (and of course a great Irish coffee).
The Guinness Storehouse (museum) also has a restaurant (buffet), but you have to buy a ticket to the museum to get to the restaurant. The rooftop pub offers a great view of Dublin.
Dublin has really a lot to do, and also consider activities like the ghost bus tour, organized pub crawls, and the various walking tours that are available.
There are numerous day tours that are practical from Dublin so you minimize the time you lose by changing places to stay. Examples are day tours to Belfast and Newgrange. If you don't want to do the day tour to Belfast then there are very frequent trains between Dublin and Belfast. Do visit the IRA museum in Belfast if you don't visit by day tour (the museum is a regular stop on day tours from Dublin) and the wall between the Catholic & Protestant parts of Belfast.
We went to Blarney Castle on one of our trips to Ireland, though it was more because my wife wanted to go to the Blarney Woolen Mills factory store. We took the train from Dublin to Cork (2 hours), then a taxi to Blarney (which is tiny), and spent two nights at the hotel connected to Blarney Woolen Mills.
It was a nice trip and the grounds of the castle were more interesting than the castle itself. Don't actually kiss the Blarney Stone (yuck).
We used a car on one trip to go up the Irish Sea coast from Dublin, but I have not visited the Atlantic coast. The only challenge was driving on the left. I recommend against driving yourself at all unless you are already familiar with driving on the left. Absolutely don't drive yourself for several days after arrival. Driving away from the airport in a strange city after an overnight flight is bad enough when you don't have to drive on the wrong side of the road.
The Temple Bar area in Dublin is well known and you should go there once. However, it is mostly tourist traps and expensive pubs with live bands that substitute volume for talent. Also lots of Europeans in bachelor/bachelorette party type groups making a conscious effort to drink themselves into unconsciousness.
There are really nice pubs all over Dublin, and I especially like the Brazen Head:
Another answer is correct the weather is unpredictable, but it will st likely be misty/rainy and cool. A water-resistant hooded jacket, a light sweater, a travel umbrella, and sturdy, water-resistant shoes were very useful each of the five times I have been to Ireland.
Buy train tickets at least 48 hours in advance via the Irish Rail website:
My son was age 20 when he visited Ireland and he was regularly asked for ID in pubs. His state driving license was accepted in all cases.
You will clear US Immigration & Customs at Dublin airport before you board your flight to the USA. It is wonderful as it mean when you arrive in the USA you simply go to the gate for your next flight (or to baggage claim if no onward flight). However, it does mean you need to be at Dublin airport and standing in line for check in three (3) full hours before your flight is scheduled to depart.
Finally, I know you are experienced travelers, but please remember to get travel medical insurance that covers medical evacuation to home. One of my friends slipped on wet rocks while visiting Ireland and injured his leg. Paying for the medical treatment was expensive, and paying for the First Class ticket to home (whole leg was in a cast) was even more expensive.
I hope you have a great trip!