Renting a house

Your question is very broad. Two things make me think a tour might be better, the ailing dad and the fact they have never been outside the US. IMHO, Ireland is best done by car, but if they mostly plan on staying put, then a larger town could work. What are they wanting to see and do?

Yes, public transportation in Ireland is nothing like other parts of Europe. The best mode of transportation is to rent a car. However, given the situation, it would be better to base yourself in either Dublin or if they want to be on the west coast, Galway, and book some of the guided bus tours! Especially one week…so a total of 7 days? Is travel time included in those days? When I go to Ireland, I count my first and last days as a wash because of travel, so if it’s 7 days, it would be a total of 5 days in Ireland. I’d potentially fly into Dublin, and look into day tours from there! Dublin itself needs two full days. There is a train, but the places you get to aren’t the most convenient and again, you usually need a car from the station. However, we did take the train from Dublin to Kilkenny and spent a night and it was easy to grab a taxi from the station into town. Same for
Dublin to Galway!

Your friend should talk to her father about what he is interested in seeing. If your friend and her husband are confident about left-hand driving, then by all means, they should rent a car. If not, they can take the bus or a train. For "heritage" sites in Sligo or Kerry, a taxi may be needed if the sites are not within reach of the bus. To this point, I suggest your friend find out as much as she can about the family's roots and, assuming that's the purpose of this trip, make those stops the priority.
For a trip of one week to Sligo and Kerry, flying into and out of Shannon seems sensible. And two stays only, one in each county. With a car, they can stay anywhere, so a broad geographic search of AirBnB and/or booking.com with appropriate filters (e.g., a ground-floor bedroom and bath for grandpa if he can't/doesn't want to climb stairs; on-site parking).
I wouldn't recommend bus tours for a person described as "ailing," as there may be days when he doesn't feel up to something like that, and if he can't go, then will the others go without him or just take the hit on the tickets?

Depends on the budget but an idea would be renting a cottage and do tours with a private driver

In Kerry we used Kerry Experience Tours for daytours and Kerry Coastal Cottages for renting a Holiday Home

The Holiday Homes are on the Ring of Kerry and Skelligs Ring most with wonderful views
We enjoyed the Ring of Kerry Tour very much

Another plug for Kerry Coastal Cottages. We've booked with them several times in the past. They offer some unique accommodations in County Kerry, including a few thatched cottages as I recall.