Compressed schedule workable?

Can you let your sister go home and without you so you can stay as long as you wanted originally?

My sister doesn't fly often and she has never done so by herself. If I told her to fly 10 hours home by herself, she probably would not go on the trip at all.

if you hire your car at the airport then you will not have to go anywhere near Edinburgh city centre with the car,very easy to get onto the road to Stirling from the airport. Buses and trams from the city centre to the airport take about 30 minutes.

On Day 5 you've got an awful lot of sights compressed together. If you are really keen to see some of the Isle of Skye and Urquhart Castle (which is right near the Loch Ness Centre & Exhibition -- not cheesy and very interesting), then I would spread those over Day 5 and 6 and skip Culloden Battlefield. OTOH if Culloden is more important (and it's right near Cawdor Castle) then eliminate something earlier in the trip such as Stirling Castle or Eilean Donan Castle or Urquhart Castle.

Distances take longer to cover in Scotland than they look on a map, to American eyes anyway.

I agree with Unclegus that the car rental facility at the airport is well connected to the main highways so you won't be navigating crowded city streets. The important thing is to ask the car rental agent (the one who actually signs you into your car) which signs to follow at the roundabout(s) leaving the airport to get on the M9 toward Stirling. Your desired direction may seem obvious to those who've been there, but to a newcomer adjusting to driving on the "wrong" side, it can throw you into a panic to find no signs matching what you expected. For example (I don't know this for a fact, just hypothetically) if there are only 2 choices, Livingston and Linlithgow, you as a tourist would not necessarily know that Linlithgow is the one en route to Stirling.

What if we saved Urquhart/Loch Ness/Culloden for the morning of day 6? Since the train from Inverness to Edinburgh is only 4 hours, we wouldn't need to catch it until mid-day. (I'm assuming there are multiple trains to take during the day.)

I'm also wondering if we need to take the train all the way to Waverly or if there is a stop closer to the airport if we decided on a hotel near there.

there is a stop at Haymarket a couple of minutes before Waverley, walk out the door and there is a bus stop and tram stop that will get you to the Airport in about 20 minutes.

You are trying to cover a lot of ground in a little time.

Personally, I would drop Inverness and spend another day on Skye. Culloden is just a field and there isn't much to see there other than taking the audio tour. When I was on Skye, the mist and cloud covered the Cullins (mountains) for two days, so I was glad I had longer to see Skye in clear weather (I was there late June).

I would also skip Loch Ness as there are many nicer lochs that you will see.

It takes longer to get from Inverness to Edinburgh by train than driving.

I had a lot more time in the area than you will so I stopped at Culloden and really enjoyed it. But, that's me, I like battlefields and the stories that go with them. It took about an hour to do the audio walking tour and the visitor center. I skipped Loch Ness, as others have said there are nicer lakes to see. If you don't have to stop in Inverness (to sleep) then I would also skip this and spend the time on Isle of Skye. I rented my car at the airport as it's easy to get out of town from there so that is a good plan.