1 week in Ireland without a car

First of all, good on you for just thinking about two locations for your week! Usually people come here with crazy long impossible lists. You’ve started out right! A car would not be necessary. There are good train and bus connections between Dublin and Galway. If you supplement with a day tour to the Cliffs you will do great.

Expect less daylight and cooler weather (dress appropriately) and you will have a great trip.

Sounds like a great plan to me. I’d split the time in favor of Dublin—Galway is a small city, and a full day, in addition to your arrival afternoon, departure morning, and evening after a day trip would be plenty of time to cover the town.

Looks like a good plan to me too.
Though you can indeed cover what there is to see in Galway in a day, adding extra time there in order to book tours of the surrounding area, to include perhaps a trip over to Inishmore on the Aran Islands, is a great idea. If that's of interest you might consider flying versus taking the ferry - might be a much smoother ride plus you'll add a scenic flight into the mix.
As was mentioned, be ready for shortened daylight hours (ie get an early start) and pack a goretex outer shell in anticipation of wet weather.

I think you could take a bus directly from Dublin airport to Galway. Taking the train requires you to get into Dublin city first.
Have an awesome time!

I think your plan makes sense, but try to keep some flexibility in the middle in case you have bad weather in Galway and want to retreat to the big city a day or two sooner. I'd think in late October you could make last-minute changes without too much trouble. Several companies run buses directly from Dublin Airport to Galway, though I haven't made the trip.

Alright, flights are officially booked!!

Changed the dates, so now we gave 8 days, late September to early October.

I've booked a hotel- Skeffington Arms for the first 3 nights in Galway. This is a budget trip so it seems to fit the bill.

Now I'm trying to figure out where our remaining nights should be spent.

I do really want to go to Belfast, the Titanic connection and the history of The Troubles really interest me. Debating if it's worth taking a 6 hr train ride there from Galway to stay overnight or if I should train to Dublin and do a daytrip to Belfast?
Another option would be to go Galway to Dublin, stay a couple nights then 1 night in Belfast and back to Dublin for a night before flying home but that doesn't seem very efficient.

Cork also looks interesting but not sure it would fit well.

Final question for now..does Giants Causeway require good walking ability and/or balance? Trying to see if I should rule that out.

Jumping in on Giant's Causeway as I have visited there with mobility issues...

There is a shuttle from the upper parking lot to the Causeway. I seem to remember scouring our pockets for change to pay for the shuttle, so you'll need to research and prepare for that if your day tour doesn't.

Once at the Causeway, there was plenty of flat terrain for me to get a great view. I used one of my hiking sticks with it's rubber tip on, and was able to walk out a bit.

Since it appears that you'll be visiting the Causeway as part of an organized tour from Belfast, you should check with the tour operator to see if there's any accommodation for those with mobility issues, ie the shuttle down to the rocks from the parking area. The path down to the bottom isn't particularly treacherous - it's a paved track easily managed on foot, though the uphill stretch on the way back may be something you'd rather avoid in favor of the shuttle. I would think that hiking poles would be overkill unless you wanted to go wandering out onto the rocks themselves, and even those aren't especially daunting unless you had balance issues or they were wet, in which case prudence would seem to dictate avoiding them entirely.
We found central Belfast to be pedestrian friendly.

Edited to add regarding the Causeway: you might want to look at photos of the shoreline at the bottom of the road to see what you think. As was mentioned, the area at the bottom is fairly level and wouldn't appear to pose any great challenge. Most of the basalt formations are pretty level too - ie not requiring any active "rock-hopping" unless you particularly want to give it a try - you'll see plenty of people doing just that.

Check out Paddy Wagon Tours. They have day trips from Dublin to Belfast and the Causeway. They also have day trips from Belfast to the Causeway. Geri, Novato CA

We were at Giant’s Causeway 2 days ago. The shuttle is 1 pound each way, and you can purchase a return ticket from the driver. We had no coins, but the driver gave us change (1 pound on a 5 pound note for 2 people, return). There is a visitor centre at the top, which has a fee and which currently must be booked online in advance. If you go through the tunnel to the right of the centre, the shuttle bus is there (looks like a normal city bus), and there is no additional entrance fee. The road down/up is quite steep. Many people walk down and take the bus up. I think it is worth the pound to take the bus down, too. There are easy pathways at the bottom. You don’t have to walk on the rocks, which are very uneven and can be slippery when it’s raining (not unlikely).

Oh, and I wouldn’t suggest trying to do the causeway, Titanic Experience, and murals/troubles tour all in one day.