This is a very excellent trip. You'll begin in some less-known sections of the park, but my favorite. Are you planning on hiking up Dawson Pass then around and down to Old Man Lake? If not, the hike up Dawson can be strenuous at times, but it is well-worth it if you feel you can do the mileage. You'll get amazing views of Two Med lake and down into the Nyack Valley.
To answer your questions---
I wouldn't worry too much about bugs that time of year. You'll cross enough varied terrain that it's possible. If you have a fully-enclosed shelter, a headnet should be enough.
I'd be surprised if you didn't run into a little snow (Gunsight Pass usually holds snow banks into August--we'll see how the weather goes this year). But it should be minimal. Trekking poles and careful footing should be more than fine so long as you know what you're doing.
Water in Glacier is usually very clear, just be mindful of finding it at a stream. It can get a big boggy if standing. I use AquaMira drops for treating my water. That's been enough for me.
If you can get East Glacier for your weather, go with that. If not Babb, MT is probably okay (St. Mary doesn't seem to show up). But I stop checking the weather after it turns in a week or so. July, August, and some of September are usually very nice up here. There are a few dreary days, but they're the exception.
Trail runners are more than fine. The Fly Creek UL is too. (Have you already bought it? If not, TarpTent has some roomier, lighter shelters which I prefer if I'm sharing with my wife after crawling in and out of the FC at a local shop.)
Fishing is really up to you. Some of the lakes that you'll be at will have some very good fishing (Gunsight in particular has a reputation for nice rainbow). Pick up the Fishing Glacier National Park book for some good advice there. Just be very mindful of bears and how you clean your fish and keep your camp. Whether or not to take it is really up to you and how much you enjoy fishing vs. other activities. I have a fly rod, but I don't usually take it backpacking (mine is only a two-piece). I do sometimes day hike with the intention of fishing. That's just how it works for me.
Let me know if you have more questions. I might see you out there.