I was just on the Summit in August on their Alaska itinerary. I took my father to celebrate his 75th birthday. I saw nothing wrong with the ship's wear that you should be concerned about. As for doing shore excursions on your own, that would depend on your comfort level. Considering the language barrier (if you don't speak the language), and being unfamiliar with the area. You will have to look at the ship's itinerary and consider how
long you have in each port. Then consider time for the ship to get cleared and disembark, and returning to the ship approximately 1/2 hour before it is scheduled to sail. Personally, I would do Capri through the ship. Especially since it is separated from the mainland. Remember if you do excursions on your own you are responsible for getting to the next port if you should miss the ship.
Congratulations on your 60th. I think the speciality restaurant would be a lovely way to celebrate. You can book a reservation before sailing. Whether you will want to book for more than one night will depend how much you value the experience and the extra cost. I find the food in the main restaurant to be perfectly fine for my taste. Celebrity provides good food, good entertainment, and excellent service. I find their crew to be exceptionally friendly.
The Mediterannean is one itinerary I have done yet, so I can't recommend any "not to be missed" experiences. However; look under ports of call and Europe Port calls, on this site and you should find comments from others that have sailed to those destinations.
Enjoy your cruise.