Kinda, sorta. It's a little convoluted.
There are 300 people allowed up Half Dome per day. 75 of those slots are going to be available only to wilderness permit holders that checked the Half Dome box on their application and whose itinerary reasonably includes it. Still a lottery though. The whole first-come, first served theory can be somewhat sketchy if you're trying to plan and counting on it. I quit caring about Half Dome a long time ago, so in case you haven't seen this, it'll explain the process better than I ever could:
http://www.nps.gov/yose/planyourvisit/hdpermits.htmIf you're coming up the Mist Trail, you're more or less going to be day hiking, and without that wilderness permit in hand you're going to get into that whole lottery system for the other 225 permits available to the thundering herds. I'll tell you though, while Half Dome is iconic and all that, Cloud's Rest is higher, has better views, and is more of a backpacking achievement, and with 4-5 days, you have lots of choices in how you get there via the backcountry. Some less crowded and more picturesque than others.
Avoiding mosquitoes and seeing waterfalls at their best don't really coincide seasonally. Peak flows are May through June, with the water tapering off substantially by mid-July, then usually being not more than trickles for many of them August through October. On the other hand, while the falls are at their best, the mosquitoes are at their worst in most of the park. If you're in the Valley or the lower elevations, it's not so bad, but it can get pretty gnarly up higher. For people that make avoiding bugs a big part of their criteria, August is the month they usually shoot for, but, so much for the falls.