Itinerary suggestions and where to stay near Dingle, Ring of Kerry

I think that most here on the forum would suggest avoiding Killarney as a base. The town has become something of a congested tourist trap, especially in June/July, and traffic getting in and out of there can be murder when the tour buses are on the move.
I'd suggest looking at the stretch between Castlemaine and Glenbeigh. There are some very nice B&B's and self-catering cottages in the countryside, and the location would be more strategic for exploring Dingle, the National Park, the Gap of Dunloe, Portmagee and the Skellig Ring - all less than an hour's drive away. That way you could unpack once for 4 or 5 nights, settle in, and enjoy a little of rural Ireland after a full day out and about.
If you'll be heading to Scotland after your 10 days in Ireland I'd strongly suggest booking your flights as an open jaw ... flying in to Dublin and then home from either Edinburgh or Glasgow, booking your car one way from Dublin with a dropoff at Shannon - Aer Lingus has at least one cheap, direct (non stop), flight to Edinburgh per day. That would save you a day by not having to backtrack to Dublin.
We're not big fans of Galway, but we love Connemara. During our visits we usually stay in or near the village of Clifden which is central for exploratory drives around the Connemara Loop, Cong (where the Hawk Walk is) and the ferry landing at Rossaveale for the day trip over to Inishmore. Some here suggest flying rather than the ferry. We've not done it that way but it does look like a great alternative to the boat trip.

Landings on Skellig Michael are weather and sea-state dependent and cancellations are not uncommon. Also, those trips need to be booked months in advance as they fill up quickly. There are also boat trips that circle the island but don't actually land on it for the hike up to the beehive huts. Those can usually be booked a day or two in advance from the Skellig Center or private operators in the harbor.

I'll offer a consenting opinion-- I too had been advised against staying in Killarney, but ultimately it worked best for our itinerary so spent 2 nights there this past July. Our hotel was The Brehon, about a 15-20 minute walk to the main streets and it was definitely the "splurge" hotel (even though it was still very reasonably priced) of the trip. Killarney was definitely crowded, but we really enjoyed our time. We spent most of the days outside of town with Ring of Kerry, Muckross House/Torc Waterfall hike, and Gap of Dunloe hike so avoided most crowds. We found a fantastic bar with live music one night and had excellent Italian food there.

I'm not saying its your best option an I know this is not a popular opinion but just thought I'd offer I was pleasantly surprised by Killarney. We also spent 2 nights in Dingle which was the highlight of the trip.

We're traveling to Ireland in October 2019 and staying in both Dingle and Kenmare. Kenmare because it's a good base for RoK, Beara Peninsula, Killarney National Park, etc. and Dingle because --- well, it's Dingle! I've got reservations at Rockcrest House B+B in Kenmare, and O'Neil's B+B in Dingle. They both fitted our price range, good location and had many good reviews.