Help with itinerary!

You don't say whether or not you will have a car, or if you are also flying out of Dublin, but I would suggest you start by planning a somewhat circular route if you have to return to Dublin. This could mean going west towards Galway first, then down south towards Killarney/Dingle, then back towards the Kinsale area, and north again, maybe to Kilkenny with the goal of ending back in Dublin. If you aren't going to rent a car you will be more limited to staying in towns that are easy to get to by train/bus, and that you will be able to get tours from to see some of the sights. If you provide more details about how you are travelling, and what kind of activities interest you (ancient sites, or nature walks for example) then you will get some great advice from people here.

We will be arriving and departing from Dublin so we will need to circle back around. We certainly can rent a car (hoping that there are no age restrictions since my husband is 74) if that would be preferable to seeing the area. Our interests are music, pubs, local scenery, cultural activities, and perhaps some easy hiking. We’d prefer hostels or bed and breakfasts over hotels, though I wouldn’t be against a stay in a castle for a night. We also would prefer to stay where we can access free internet. Recommendations for pubs or restaurants (no fancy places) would be helpful too.

If you want to stay in hostels then you should check out the hostelling international site and figure out which towns have hostels, and read the reviews and decide if you actually would want to stay in them. Dublin, Galway, Killarney all have hostels, some more than one, but many other small towns won't have a hostel. Some hostels have private rooms, some only have dorm rooms with bunk beds, and some dorm rooms are mixed gender, also. I found that hostels varied in quality quite a bit, and some didn't really resemble their online photos so it's best to read recent reviews. I stayed in a hostel in Kilkenny where the showers were so dirty I wouldn't use them, but most of the others I stayed at were passable. Having a car will make it easier and quicker for you to get around, but it will also add to the cost. If you don't have a car you have to plan your travel well or you will miss seeing a lot of things because you will be on a train or a bus trying to get to the next town. Without the car it is best to base yourself out of one town for at least two days (four is better) so you have time to do guided day trips to locations you can't get to on public transit. Since you are leaving quite soon, and you haven't booked a car rental or any accommodation then I think the first decision you need to make is how you are going to get around. You may have difficulty booking both B&Bs or hostels at this late date in the more popular locations, but if you have a car you will have more flexibility to stay in smaller towns, or outside of the popular areas.

With respect to leisure time and sitting in pubs......we had the same objective and the pub we enjoyed the most in our 12 day tour of Ireland was the Poets Corner bar in the Old Ground hotel in Ennis....we met some of the nicest people on the planet in there and had some uproariously great and fun conversations. We also enjoyed the "street scene" in old town Killarney...bag pipe players and street musicians and just a pleasant relaxed atmosphere....tons of nice pubs....Murphy's Bar and the Famous Danny Manns were great fun.
If you want to stay a night in a Castle I will recommend the Dromoland near Bunratty (ten miles or so from Shannon airport).....a bit expensive but as a "once in a lifetime experience" it's darn well worth it. The Mount Juliet Estates just outside Kilkenny is, technically, a manor house but the difference was lost on us.....it was the most relaxing and pleasant time we spent while in Ireland and we're agreed we will go back there.
The age limit (if it's enforced) for renting a car is 75 so you're just under the wire. If you decide to rent a car be sure and have an excellent Nav system and please stay off any road designated with the letter "L"....those are about the width of your driveway. The motorways designated by the letter "M" are excellent and the "N" roads are typically not too bad but can be quite narrow in places. Also, be certain to understand what you're getting for what you spend on insurance. We learned before we left that if you get only the "standard" insurance of CDW and Theft you'll have a 2500 euro deductible. You can make that go away by purchasing the Super Coverage (SC) which I did. After arriving in Ireland, I learned that wheel covers; tires and mirrors are not covered at all by insurance. We've decided we'll hire private drivers for our next trip to Ireland.

Our favorite traditional music was in Doolin. Nice pubs there, and good beer.

On the Slea Head Drive, we stopped in at a pub where the patrons were local and speaking Gaelic. THAT was authentic! Would have been neat to stay off the beaten path and go to a few of those pubs at night. Dingle itself had interesting pubs and music as well.

I seriously doubt you can go wrong wherever you choose to roam. People in Ireland are very friendly and ready to chat over a pint! We actually had a heck of a good time bumming around pubs in Dublin, too.

Milky Gallivan's and the Brazen Head are both well-reviewed for putting on a great traditional show in the evening. We stopped at each while during the daytime but didn't see a show at either so I can't personally comment.

We particularly enjoyed Muckross House and a walking tour with John Connolly (the Burren). Both were insightful to traditional Irish life over the years. Also, I would HIGHLY recommend a visit to the Blasket Island visitor's centre. It was a moving experience and made me wish we would have had time for a visit out to the islands themselves.

Other thoughts? Ireland is full of treasures around every corner! If you have the time, I recommend a leisurely pace so you can stop at wherever tickles your fancy. :)

We stayed at B&Bs everywhere on our trip. Great places and friendly hosts. It was fun to chat with other guests, too. Our favorite B&B was Friar's Glen in Killarney. Best B&B I've ever had the pleasure of staying.

Your plan to pub sit and meet the locals is the best way to experience Ireland, IMHO. I can recommend a few accommodations to consider.
Dublin: Buswells Hotel near Trinity and Parliament. Good restaurants, shopping and sites in walking distance.
Kilkenny area: Lawcus Farm Guest House, just outside the village of Thomaston. Lots to do nearby, though you will likely want to stay in to visit with Mark and Anne Marie, the hosts!
Kinsale: The Cloisters B&B, run by Orla and Aileen. Just up the hill from the harbor and downtown area. Truly a lovely village on the water.
Dingle: Pax Guest House, up the hill from town overlooking the harbor. John sets a very high standard for B&B hosts.
Happy travels!