Hi there pevelg! Welcome to the forum.
Sounds like a nice trip you've got planned, getting back to your home stomping grounds. Good for you. Let me try to answer some of your questions the best I can.
First, your 5 wt rod will be perfect for Montana in the summer (assuming it's at least 8'6" long). While Montana may be a special place, you do not need any special fishing gear, you'll be fine with what you have.
Early August is never going to be the best fishing conditions here. This year most of the rivers in the Missoula area peaked a month early, around May 14, which is not a good thing. Fortunately we've had a very wet May and beginning of June so while run-off was over really early, the flows have stayed up and consistent, though below average. Sorry, I get a little obsessed with river flows and fluctuations...what this means is that by August in a normal year the water is low and beginning to warm. This year could be a little more extreme than the normal years. When the waters start to warm, focus your fishing in the evening into the night when water temps are dropping.
Check the flows here. USGS Current Conditions for Montana_ Streamflow
Scroll down to the Upper Columbia Basin to find the rivers in the Missoula area and the Yellowstone Basin for rivers in the SW and Bozeman area. These charts show the historic average and the current daily flow for comparison.
On to the fun stuff. The Bitterroot is great river, for reasons already discussed the upper portions of the rivers is where you will want to be, in general. The lower rivers, particularly the Root will warm a lot faster and the fishing will be bad. The West Fork Bitterroot comes out of a large reservoir and flows ice freaking cold all summer. It will be a really good place in August. It is 70 miles S of Lolo though. The creeks could be good options as trout seek solace in the colder waters. Lolo is a fun one to explore.
On your way from Missoula to YNP you will pass through basically some of the best trout fishing country in the world. You could swing down through the Big Hole and make your way to the park via Ennis (Big Hole, Beaverhead, Madison) or cruise straight to Bozeman via 90 and then go down to the park (Gallatin, Yellowstone).
It will pretty much be a Hopper, terrestrials, caddis at night game by then.