I love June. The earlier in the month you can go the better.
I use the neck cooling bandanas to keep cool but plenty of people use fan misters. I figure I get sprayed enough from people who use them but don't aim well so I don't need one of my own.
The rain: yes it does rain nearly every afternoon the week we go but we just put on our ponchos and keep on going. Alot of people leave so the lines get shorter for everything.
The lines: We use fastpasses alot and follow TGM's best days for which park and it worked great. The only day on our last June trip that we didn't follow his plan, it was crazy. The lines can get long from about 11am on so you just have to plan for that.
The food: If you are doing CS, we found it was always better to eat early. The lines for CS food stayed busy all day but if we got there right when places opened, we had hot quick food and plenty of choices of where to sit and eat.
Star Wars Weekends: If you go to DHS on those weekends, they will be crowded. But if you want to see the special 10am Star Wars parade and do the other special Star Wars stuff, it's the only way you can do it. We planned our day carefully, got fastpasses and had a great time.
I love going in June and we'll be going in June 09. We just couldn't go as early in the month as I normally like but it'll still be fun anyway.