7 days in Ireland from Shannon to Dublin - first time visitors

I would strongly suggest that on your arrival in Dublin, you turn in your car and use public transportation during your visit. Parking is scarce and expensive, and traffic is horrible. Anywhere you wanted to go outside the city is easily reachable by bus or a short tour. There is no sense in paying for two or three extra days of rental only to leave it in expensive parking.

I suggest that you also see Kilmainham Gaol while in Dublin. It is an important site in Irish history. You might also consider a hop on hop off bus tour to get your bearings. The narration is fun, and you can use it for transportation for the length of the ticket (24 or 48 hours). The whole circuit takes about 90 minutes.

I'm curious where you are flying from, since overnight flights from the US usually arrive in the early morning.

Sounds good to me!
I have never been to Bunratty Castle, something to check out.
On day 7, I would recommend going to the following:
Monasterboice, New Grange, and Trim.
Have a wonderful adventure!

I agree drop the car when you get to Dublin, preferably at the airport. Why pay for the car for three more days in an already expensive city, when it will just be sitting? In County Clare consider staying in Ennis, very authentic town with good Irish music. Can't say enough good things about this town. (We stayed at the Grey Gables B&B in 2002 and again in 2016). Dromoland Castle, which has been repurposed as a posh hotel, is very worth a visit, have lunch in the bar which has nice atmosphere and courteous service. Nice gardens and gift shop at Dromoland, too. Dublin is good for two full days, maybe three. I think your day three looks very ambitious. The Aran islands probably deserve a full day, if you are going to go there at all. I think it's very worthwhile but have never been myself--so much to see in Ireland, the islands always seemed like an advanced topic that were too hard to fit in a two week itinerary. Also put off by the fact that it's very weather dependent and the sea off of Doolin was crazy rough both times that I was there. Trim and the passage tombs are a worthy visit with more time, we visited last May. I was a little disappointed by the fact that so much is a reconstruction and they don't know how it originally looked at all. The town of Trim is extremely picturesque, some excellent hiking trails around there. We missed out on the castle keep interior, which would have been a highlight, because the tour was not offered until an hour after we had to leave (can't visit on your own). Generally, expect things to take longer than you expect them to take in Ireland.

II love the folk park at bunratty!

Overall this looks like a reasonable itinerary that will allow you to enjoy the area except for Day 3 see Cliffs of Moher; the Burren; ferry to Aran Islands. If you’re spending the night in Doolin, I believe you can take a ferry to the Aran Islands (I went from Galway), but it will take up most of your day to tour the island and enjoy it. Usually, you depart the island at 4 or 5 p.m. So you may need to view the Cliffs the morning of the next day. They are spectacular, and you can spend hours walking along them to enjoy the views, or you can just walk a short distance and still see magnificent scenery.

For Dublin, I also recommend the Kilmainham Gaol, and the tour at the National Post Office; both offer an outstanding perspective on the history of the Irish struggle for independence. You could tour both on Day 7. Get tickets for the Gaol early. Allow some time to walk around the city along the areas near the river. You can find good music in the evenings as well.