Sorry, was away from my computer for a bit, hence the late reply.
Weather. In all honesty, guessing Scottish weather is like guessing the lottery numbers. You could go in August to avoid the rain and it would tip it down and September would be lovely, or it could totally go the other way. If you're used to it and bring layers with you there won't be any problems. I'd be more inclined to book your time based on what you want to do rather than taking a punt on the weather. I've looked at my pictures from September last year and it looked fine, but who knows!
Football. I'm not a fan of the Old Firm, but you should be OK with getting a ticket for all games apart from a Rangers v Celtic game. Parkhead is the nicer stadium (and probably the better football), but i totally get it for Defoe/Gerrard. I'd be tempted to contact the box office in advance and see what help they can give you. One thing to bear in mind is that you can't drink at Scottish football grounds as they don't trust us!
Planning. The elephant in the room for your timing is the Edinburgh festival, that runs for the whole of August. Edinburgh will be heaving, but you could always take a trip from Glasgow for the day or stay in Stirling etc. It's absolutely not essential, but it could be something to consider. Orkney isn't really a day trip - it would take you a day to get there, and you need a couple of days to explore. Skye is a much better option, as you can drive there - I'd hire a car from Glasgow rather than Inverness. When my wife was first over from Australia we did a trip that started from Glasgow, drove through Loch Lomond, and drove to Skye through Mallaig. From there we then drove past Loch Ness to Inverness. From Inverness, you could then do a little bit of the North Coast 500 (this is a driving trip over a week across the top of Scotland - super, perhaps too, popular, but it's popular for a reason) or drive down to Aberdeen or Dundee or Perth, then back to Glasgow. The good thing about the central belt is that everything is so close - Glasgow to Edinburgh is 1 hour away, so is Perth, and Aberdeen is a couple - for a North American these distances are nothing.
Others. For accommodation, don't discount Youth Hostels, especially up north. You can totally book private double rooms, and they are decently priced. They are also handy for organising tours etc. on your behalf. Airbnb and booking.com are also good here. In August you might also be able to book empty student accommodation - look at Glasgow and Strathclyde uni to check. For music, Glasgow totally pumps Edinburgh. We have the better venues, and a lot more bands play here. If a band you like is playing the Barrowlands Ballroom or King Tuts don't hesitate to book a ticket. Look at Tickets Scotland, Gigs in Scotland or Ticketmaster for details. For other stuff, look at The List and The Skinny magazines online, which both have Glasgow an Edinburgh listings. Can hit you up with restaurant tips for Glasgow, lots to choose from.
Off the top of my head, here's a suggested plan for 7 days, driving from Glasgow: Glasgow 2 nights>1 day drive to Skye via Loch Lomond. 2 nights in Skye.>1 day drive to Inverness>2 nights Inverness>Drive to Perth or Dundee via Pitlochry>1 night, drive back to Glasgow. Full on, you'll be knackered, but you'd see loads. If that's looking tight, skip the overnight in Perth/Dundee and just drive through.
By Train. Glasgow 2 nights or 3. Early train Glasgow to Edinburgh; day in Edinburgh & late afternoon train to Inverness. 3 days Inverness with day trips (day trip to Skye is harder here) around. Train back to Glasgow, with maybe a stop in Dundee or Perth. You can do day trips to Skye from Glasgow (
Isle of Skye, Loch Ness & The Highlands Coach Tours from Glasgow - only found via Google, no idea if they are good), which means you maybe use Glasgow as a hub and don't bother with a car.
If you need any more help or need me to book anything don't hesitate to get in touch.