Traveling with my mother for the 1st time to Ireland in July

Is this rapid change from place to place a typical and preferred travel style for your family? It would be too many places for too short of a time for many people, but if that is the way you are used to traveling it can work out fine for you.

I think you already have a sense that this may be too much and are thinking of how to "whittle down." What areas of Ireland are special to your family history? Is there an area that you know was the area where your ancestors live?

Some who have been the Rock of Cashel more recently may be able to better advise, but it is my recollection (from a trip about 5 years ago), that it is a rather steep hill to climb from the parking lot. I don't recall any special-access parking for those with walking difficulties, but you could email ahead to check. If so, perhaps you could drop your mom off and then park the car to meet her.

Driving through The Burren would be easy, but for much more exploring, it might be difficult for your mom to walk on a lot of the uneven rocks, but you will be a better judge of that.

Do you know from which city or townsland your mom's family is from?

The distances take much longer than you are planning. Double your drive times. The roads are often narrow and winding. You miss the best of Ireland if you stay on the highways.
I sense that your mom is older since she has mobility issues. Slow down so she can enjoy the trip. I’d go to the area her ancestors are from. Then build your trip around that. You won’t see as much as you have listed but that can be for another trip. Pull out a (paper) map and a guide book. Sit down with your mom to plan your trip.
I’d stay out of the tourist areas. Stop often at local pubs and wherever your muse leads you. No Rock of Cashel, no Bunratty Castle, no Blarney Castle. Your mom won’t be able to walk the Cliffs of Moher but there are other beach areas she can view. You can see the Burren from the car but it takes walking around the rocks and monoliths to appreciate the area.

Very exciting trip ahead of you!! I took my first trip with my daughter in 2017. It was stunning and, for me, life changing. I am heading back for a solo trip later this year!!
But I must agree with some other replies that you have an extremely ambitious itinerary planned. On our journey, we spent nine days driving (about 1100 km) and barely got out of Co. Clare!!! What we did spend the time doing made for wonderful, impactful memories. For us, seeing the non-touristy Ireland was most important. Along those lines, I would encourage you to give yourself a bit more time at a smaller selection of places you really look forward to seeing.
The countryside is so worth enjoying. One of our most amazing visual finds was discovered by surprise when we pulled onto a side road to let traffic go by.
As Rick Steves says in his Ireland 2018 guidebook, “Dont overdo it - few travelers wish they’d hurried more.” Safe travels to you both. I hope your first visit to Ireland will be as memorable as mine was!

You’re doing a lot that first day. You will be more tired than you might think, especially if your mother is older. My mom is in her 70s, and she really didn’t have the ability to do more than one stop before dinner and an early bedtime. Consider adding a second day for Dublin.

We enjoyed shopping at Kilkenny, but the castle wasn’t much to write home about, except for the shopping in the old stable block across the street (Irish artisan items—got lots of stuff there!). For interesting sites, I would go to the Rock of Cashel instead. A fun stop in the Wicklow Mountains is the Avoca Woolen Mill—make sure you go to the actual mill with the factory tour, not the large store several miles away. The shop at the mill is sufficient, even if it doesn’t have everything the large store does.

For the Dingle Peninsula, I really recommend getting a local tour. That way, neither of you has to worry about driving on the very narrow, winding roads, and you can both enjoy the scenery. We had one recommended by our B&B owner when we found the one the Rick Steves recommends was already full. He was a terrific driver/guide, a retired school principal who took about twenty of us in his small bus to all the sites on the way. He promised us 2 1/2 hours, and it ended up 4 1/2! IIRC, it was about 60-70€ each, not including the pretty substantial tip we included for the extra service/time.

I believe you can drive right up the hill to the entrance of the Rock of Cashel before driving back down to the parking lot to drop off your mother, and you will need the wheelchair at the Cliffs of Moher to get the long walk from the parking lot.

You're doing too much! July 12th looks especially crowded. Muckross House and the Torc will take half a day- include driving from Kinsale and over to Kenmare and that's about all you're going to get in that day. And you want a few evening hours to casually stroll Kenmare.

Rock of Cashel is worth seeing, but you probably won't spend a lot of time there, so I think you could do both Kilenny and ROK in one day.

Overall, I don't think it's terrible to have an overly ambitious itinerary, but be prepared to let some things go. On my first trip I didn't get to see the cliffs, but now I'm going again! Have a great time.