Sawtooths July 5th thru 12th

Definitely Black bears, but not likely to be a big deal. Hang your food and you'll be fine.

Skeeters will be out. How bad depend on your route and camp locations. If you plan on spending a night or two down in the thickly wooded valleys, or at one of the lower lakes, a full net will be welcome. If you plan to camp high and in more open terrain you'll most likely not notice them.

You will not have to deal with bears in the Sawtooths. There are maybe two spots in the entire range that have bears on a regular basis, and if you are there, you are seriously lost and out of touch with why you came here.
Bugs now are another matter entirely. The water is high, but still going to be an early year for snow runoff. This years water was at 71% of normal. But its been a cold spring and so its coming off about normal.
Drop me a line and I can give you specific details about anywhere in the range and hopefully add some highlights to your trip.
have fun!