Re: August or September
Personally, I think if you're interested in a particular festival, then by all means try to go when it's happening (and book SOON!). But if you just want to do the everyday sightseeing sort of stuff in Edinburgh -- and if budget is an issue at all -- I'd go in September instead. It will be more affordable, more relaxing, less crowded.
Sorry not to have a hotel recommendation for you; I was younger when I was there and stayed in hostels.
Enjoy your time there. It's a beautiful city!
"I haven't been everywhere yet, but it's on my list." -- Susan Sontag