Caroline, you'll have to get your Spain-specific info from others, because I haven't been there since global warming kicked in. I hope you'll learn that you can count on air-conditioned trains, inter-city buses, restaurants, etc.
I traveled extensively in Europe last year, visiting Italy (including Sicily), eastern Germany and the Balkans. The middle 3 months of the trip were extremely hot (as in highs from the 80s to mid-90s F) almost every day. It was brutal. But I live near you, I grew up in the southeast, and I survived. Go, but drink a lot of water!
Unless you're taking a day-trip, you'll have access to your hotel or apartment in the middle of the day for cooling off. Look carefully at the city temperature trends and play the odds. Barcelona can be a lot cooler than Seville, though it might not happen in the middle of July.
My most uncomfortable times were in a few cities where either I couldn't find an affordable air-conditioned 3-star room (trying to adhere to a budget) or the air conditioning was malfunctioning. The promised a/c did not materialize on two occasions. One of those hotels was in Germany, so you just never know.
Once I reached northern Italy and points north, and in the Balkans, I encountered quite a lot of casual restaurants with no a/c. When that overlapped with having no a/c in my hotel, or when I was on a day-trip, it was hard to escape from the daytime heat. Those were the toughest days for me. I was often stuck eating inside restaurants where there was no air moving, because smoking was allowed outside. Frequently there was a decent breeze under the outdoor umbrellas, but I couldn't take advantage of it. I also hit a few rail cars on slow trains that weren't air conditioned. No problem on inter-city buses except one occasion; the bus line tried, but failed, to remedy the situation and we had a long, hot ride from Sicily most of the way to Puglia.
In my experience, museums containing paintings were always kept comfortable, but you can't count on a/c in other types of museums, especially the smaller ones.
