You sound similar to DH and myself. We life food/drink, trying local foods, walking around, exploring etc and also can't stand it too hot. These trips were pre-covid, so hotels etc might have changed.
Sicily- I think we went in Oct, but May/Jun likely wouldn't be horrendously hot. We had 2 weeks, but in 10days you'd certainly get to travel about without being rushed.
We flew in Palermo and stayed here https://extra-bed-and-breakfast.hotel-palermo-it.com/en/#main It was across the road from the airport bus stop, so very easy to get to/from. Its run by a retired couple and breakfast included the most delicious home made jams, cakes among other things.
We then got a train to Agrigento. There are massive, acropolis type ruins there, but majority of the artifacts are in the museum. We only stayed 2 days here and that was enough to see what we wanted. We stayed here, and it was about 30m from the train station, so again, very easy to get to. https://www.agoda.com/b-b-la-dolce-vita-luxury-house/hotel/agrigento-it.html?cid=1844104&ds=lSqZnvb%2BgT29oeWw It had the most ornate, over the top room decor I've ever seen. Breakfast was delicious and we'd sit on the balcony having it.
We then caught a bus to Catania. I wouldn't recommend the hotel we stayed in, as it was right next to an all night, nightclub! We did a day trip via bus to Syracuse which we enjoyed. We would have gone to Taormina, but Etna was rumbling and it was advised not to travel there.
We then caught a bus back to Palermo, but you can fly out of Catania too.
Mainland Italy
Depending what you want to see do, we flew into Naples and stayed here https://www.correra.it/en/ Caught the circumvesuvian train to both Pompeii and Herculaneum. Went to the museum in Naples where all the artifacts are. I think we stayed 3 days, then caught a fast train to Rome for the rest of the time we were there.
Northern Spain won't be too hot at that time.
We flew into Barcelona and stayed about 3 days. Then got a train to Zaragoza and stayed here www.hotelsauce.com/en/
DH had a conference there, otherwise, I'd never heard of it. There is a river, massive basilica, lots of quirky museums and things to see/do. We then got a train to Madrid.
I haven't been, but planning a trip the Bilboa and San sebastian.