I've stayed at the Comfort Hotel in Kochi: it's OK, not great. There's an interesting fire escape: a box with a plastic tube inside, throw the tube out the window, then jump inside. Hope you don't need to use it.

You probably won't have much choice where you go if you visit the Iya valley by public transport or tour, but I expect they'll take you to the two vine bridges. Both are good, and they are different. Near one is the "scarecrow village"; it's worth looking up its history and that of the woman who made them as it says a lot about rural Japan. The view from the "peeing boy" statue (and all along that road) is gut wrenchingly awesome.

The castle in Kochi is very good (and one of the original 12). I didn't have time to visit the beach but Katsurahama gets a good write up, as does Cape Ashizuri in Tosashimizu.

How are you getting to Matsuyama? The slow train across to Uwajima is the best I've taken in Japan. Very remote and scenic. Just a single car from Kubokawa and the one I rode was fitted out as a train museum. Uwajima castle is very good--another of the original 12 but a very steep climb. The line from there to Imabari is also scenic, and I enjoyed a stop off at Iyo-Ozu castle (not original, but good).

In Kochi there is a tourist bus that goes to the Makino Botanical Garden, Chikurinji, one of the Pilgrimage temples, and down to the beach area...

http://www.japan-guide.com/e/e5603.html

Also there is an train pass for Shikoku - you will need to do some math to see if it works for you:
http://www.japan-guide.com/e/e2357_shikoku.html

Thursday, no I don't get it either. Kamakura has plenty of top tier sights to see whereas Yokahama has sights one could visit if one ends up there but nothing to really pull you to it. We enjoyed about an hour to see China Town which included one of the temples and wondering through several streets but didn't visit all three main temples, admittedly, but we couldn't have spent more than 2 hours there even if we had.