We were in Varenna for 2 nights week before last (May 1 and 2, 2016) and stayed at Albergo Milano. In my opinion, the best thing about it was the view from the room (we had a very small balcony facing the lake). Honestly, I am not sure that I would stay there again. The room and bathroom were very small and the service was not very friendly.
The food in the restaurant (La Vista) was very good (and pricey). I would eat there again, but would definitely get the 3 or 4 course fixed price option. I had eaten a late lunch and wasn't very hungry so I got the Lake Como fish soup - it was delicious, but there was only about 1/2 cup of soup for $14. My husband ordered the 4 course option, and he was happy with his selection. The outside terrace overlooking the lake is beautiful, but it was closed due to rainy weather while we were there.
The breakfast which was included was good, but the service was horrible. There were coffee urns, but my husband insisted that it was instant coffee. The first morning we were not offered espresso or cappuccino, and I didn't see anyone else offered any. The second morning I did see someone getting espresso, and we were finally able to get the servers attention to get some. Getting a second espresso was nearly impossible. We went to another restaurant our second night there and was also very pleased. It was just a few steps away from the hotel and the food was very good - Osteria Quatro Pass. We had lunch at Albergo Del Sole, and it was also very good.
I am not sure about the air conditioning in late June. Hopefully someone else can answer that question.
In Varenna we went to the Villa Monastero (which I loved!) and the gardens of the Hotel Villa Cipressi (nice, but preferred Villa Monastero). We took the ferry to Bellagio and walked around the town and then went to i Giardini di Villa Melzi - a highlight of my entire trip! If we had more time, I would have visited the other lake towns. I would have loved to take the Bernina Express!
Hope this helps a little.
Kathy