Please help with itinerary

How many days total?
Since you are going to the Tatoo, you know The Fringe is happening right? I would spend at least a day at Fringe events, which I've attended a few times, and intend to return, as I find it so much fun.
The haunted tours are worthwhile , and the Scottish Museum, also.
The St Andrews Old Course is closed for golf on Sundays, and several locals conduct very colorful tours on the course that day. You can walk the course on your own, but the local tours guides have great stories.
Are you playing golf as a single?
Skye needs three nights, as it's a lot of slow driving.
Have a great time developing your itinerary.
You will have a wonderful trip!

It sounds like you have about 10 days.

To be brutally frank, I don't see how you can squeeze in so many things in 10 days.

You're talking 2 nights Edinburgh, 2 nights on Skye, 2 nights Inverness, 2 nights Glen Coe, 2 nights Oban, and possibly 2 nights in Fife. Plus a stop in Stirling and a day of golf. And you will have some long days of travel. It just doesn't add up no matter how you slice it. The distances you're looking to cover are not short either.

You need to rethink this and come up with a shorter list of sights.

If you'd like my opinion on the top options from your list, I'd be happy to share, but I don't think I can put myself in your shoes and guess what you would like most.

The first thing to do is to do a one way car hire Edinburgh to Glasgow, then take a morning flight on 23 August GLA-DUB (usually one about 0700). That then maximises your time on 22 August-
So 22 August could then be Oban Distillery in the morning, then drive down to GLA via Kilmartin.
That would be 21 August for the boat tour from Oban- which will be a full day.

13 August- arrival day
14 August- tour Edinburgh, and Tattoo
15 August- St Andrew's then end the day at Stirling
16 August- Stirling Castle at opening time up then up the A9 for Culloden and Clava, Inverness overnight
17 August- Drive to Skye via Loch Ness, Eilean Donan and the Bridge- 3 nights Skye
20 August- To Oban via the Mallaig ferry and Glencoe

then as above onward to GLA and DUB.

Either Open Jaw it (in to EDI, back from DUB) or get an early flight on 24 Aug DUB-EDI, for the return EDI to USA- Open Jaw to fly back direct from DUB would be far preferable, by a country mile.

This is rather faster paced than ideal, it really needs two of you to share the driving (especially on 17 and 20 August- 20 August is an early start to catch the 1st or 2nd ferry), and is rather shoehorning in things, but gets in everything you want.
You wanted at least another 3 or 4 days, but if that is all the time you have, then this outline seems to do the job.

Thank you all! I respect everyone's opinion and I know this seems a bit rushed, but I may never get back to Scotland, so I want to see as much as possible. We do have two drivers, so that will help. isn31c--this was SO helpful. Your itinerary and the order of things makes so much sense. I also appreciate the suggestion for 3 nights on Skye, which is doable. And yes, I do know that the Fringe festival will also be taking place at that time, and we will look into possible events we may want to book. Thank you, again!

I agree you need at least 3 nights on Skye; we spent 4 nights on Skye which was perfect but you don't have room for an extra night. We followed the On The Luce roadtrip itinerary for Skye, which is 3 days. But you can tailor this roadtrip to suit your needs and timeframe. I'll give you the link because I think this itinerary helps you organize your day, and thus your driving, into a logical route so that you are not backtracking. And you can omit anything that either doesn't interest you or you don't have time for.

https://www.ontheluce.com/isle-of-skye-itinerary/

The Glen Etive drive is spectacular! We loved every place we visited in Scotland; however, Glen Coe and the Isle of Skye are our two top favorites for sheer beauty. Just stunning!

https://www.ontheluce.com/glen-etive-road-scotland/

I'll toss in a recommendation to stay in Nairn, about half an hour east of Inverness and near Clava Cairns and Cawdor Castle. There are a couple of golf courses near the town. It has many B&Bs; I can recommend Tali Ayer.

Kilmartin Glen is out of the way vs. the other destinations on your list. We stayed in Inveraray (can recommend Brambles) and drove to Kilmartin to see the prehistoric remains. Note there are also lots of stone carvings from the historic era (as opposed to prehistoric).

I'll disagree with many others about the Isle of Skye. We enjoyed it, but wouldn't have necessarily wanted to spend more than a day (two nights) there.

I've heard many friends express regret that they went to Scotland and tried to go too many places too fast. I'd encourage you to consider being in Scotland rather than driving through it.

One other place that is known for its golfing is Cruden Bay, which is a small village on the north coast of the Bay of Cruden in Aberdeenshire. It's beautiful there and is within walking distance of the ruins of Slains Castle and the amazing Bullers of Buchan is nearby. I spend one night here and would love to come back (and I'm not a golfer). The golf course is evidently very popular with golfers (and not nearly as difficult to get into as St. Andrews). :-)