Buy a BritRail pass and an Oyster card?

For London you'll want the Oyster card, a rail pass won't work.

Outside London, I suggest you do some comparison pricing before you invest in a pass. Use the National Rail website to find trains you'll want to take. I don't think there's any train service to the Isle of Skye, you'll have to rent a car or take some kind of tour. I expect you'll find that buying individual tickets will cost less than a pass, especially if you can commit to advance purchase on specific trains. Learn more about UK train travel from The Man in Seat 61.

Does the Flying Scotsman train from London to Edinburgh

What exactly do you mean? Do you mean the restored steam locomotive which occasionally makes makes charter trips between points in UK or do you mean the frequent scheduled train from London Kings Cross station to Waverley Station in Edinburgh?

Has the Flying Scotsman locomotive got a charter scheduled to Edinburgh coming up? When? If it has, it won't be Britrail that you will be using.

You've mentioned 7 rail trips on (presumably) different days so your closest-fitting BritRail Pass is for 8 travel days within a month at $398 adult price, 2nd class. The next-cheapest option covers only your 4 longest travel days within a month at $278 per person, no 6- or 7-day version. (Discounts available for youth, seniors, kids, groups of 3 or more adults, or off-season travel, if you qualify for any of those.)

British trains between cities are operated by many separate companies, but you rarely have to worry about that, unless looking at a small "Heritage" railway.
* Passes cover the Gatwick Express on one of your counted travel days, or it you've chosen a shorter pass, then that's one of your cheaper tickets to buy online or in the station for £18-20.
* You can use the pass to make a daytrip outside of London by train (if you're looking to use the eighth day on a rail pass) but would not use it for anything within the city.
* You'd be covered by train from Inverness to Kyle of Lochalsh, for instance (your other shortest train ride), and from Mallaig to Fort William to Glasgow. I see some maps in Rick's Scotland book that indicate a Skye driving route from Portree and make it look like a train line; but it's not a train line.

I don't prefer booking all my tickets ahead if I don't have to, so I'd choose the pass for that reason. Without a pass, you would probably lock in at least a few longer rides to get Advance-rate ticket discounts.