We'll be there June 6-11 in what's becoming an annual trip due to business (followed by monkey business!). Being someone who primarily goes in the second week of June and/or the fall value season (late September or late October), the weather's the same as September (hot and afternoon rains, but manageable), the crowds are heavier than September, but far, far better than July (20 minute waits max, usually), as long as you go after the Memorial Day weekend and before around June 15th.
As for Gay Days, here's an amazing thought: In September, I think there might actually be gay people who aren't wearing red. My wife and I will wear red shirts, green shirts, blue shirts, and pink shirts while there. Gay Day participants all wear red as a sign of pride and solidarity. Since red is one of my favorite colors, I've worn it at least once two years in a row and have failed to be propositioned, harrassed, or otherwise psychologically disturbed. It's just not a big deal.
In all seriousness, the affects I've noticed from Gay Days is A) the celebrants tend to be younger and younger=louder, regardless of one's orientation; and B) Friday and Saturday night (June 6-7), if my memory serves me, will be berserk at PI. Sort of like a contained Marti Gras. It's actually fun to watch, but dangerous to those with intense crowd allergies (me).