We enjoyed the
Wheatsheaf in Swinton (near the Scottish/English border, close to Coldstream) when we were there two years ago. We found comfortable accommodation, a warm welcome, and wonderful food. Swinton is a small village, but was a good place for us for daytripping for a couple of days in that area. We enjoyed it so much that we had booked to return for our 25th wedding anniversary (in fact, I should be there now!). Unfortunately, EasyJet cancelled our flight, which effectively cancelled our trip.

The Wheatsheaf was also mentioned in a Spring 2004 (??) issue of Bon Appetit, the special issue on Scotland. You can get B&B, or dinner + B&B. We took the dinner + B&B option, enjoyed it immensely, and found that our breakfast and dinner left no need for other meals during the day!
We traveled to
Lindisfarne and Bamburgh Castle from Swinton, among other sights.
Here is a link to other castles in the Borders, though not all may be desirable commuting distance. The main
Discover the Borders site has lots of other good information. If you go to Lindisfarne, you will need to check the tide tables ahead of time, as the causeway to the island is only open during low tide, and you don't want to miss your "window" for arriving/leaving. We were able to do all we wanted to in a morning there, though it was perhaps not as leisurely as we might have liked.
I would certainly recommend a rental car for your time outside of Edinburgh. We also toured the
Border Abbeys (Dryburgh, Kelso, Melrose, Jedburgh) on a different trip - well worth seeing if you like ruined abbeys. As for getting from Glasgow to Edinburgh, I have no first-hand experience with transfers (we rented a car at Glasgow/Prestwich and drove, but of course you won't want a car in the city), but you might find the information for
Scottish Rail or
Glasgow Airport to be helpful.
Glad you took to plunge to get away on the spur of the moment. Best wishes for a wonderful trip and birthday celebration!