Looks like you have timetabled the driving but forgot that each of these places takes time to see. For example Hadrian's wall cannot be seen from the car. Give yourself two hours to go round the museum and perhaps another hour or two to hike along the wall itself. Same goes for the other locations you plan to see such as Skye: "Trotternish Peninsula, Dunvegan Castle, Cuillin Hills, and Talisker Distillery." Each of the things you mentioned are a day in themselves. Particularly how do you see the Cuillins without getting on your boots and hiking several hours up the glen.
Another factor you have omitted is the weather. Scotland gets a lot of rain. You should plan a schedule that allows for some indoor activities (e.g. museums and shopping) to be inserted when the weather isn't so good. You might want to completely rethink your itinerary based on the weather. For example Take a look at the weather for this week (mid June) to see what I mean:
Based on this week, you would want to do Skye on Monday. However, this was taken from Google and it's just not precise enough. It's not going to rain all day every day. Only parts of the day will be wet, other parts dry. You should look at the BBC website for accurate hour-by-hour forecasts to help you schedule so you can get indoors before the rain comes on, then get out again once the rain stops.
Then there's the driving. Roads on the west coast, such as near Skye, are small, twisty and require care to drive. You may find that journeys take longer than expected.
To improve your itinerary, reduce the distances you drive and give yourself more walking time at each location. Don't try to circumnavigate the entire country. Focus on part of the country but try to get a taste of each aspect.
What aspects should you cover - not places? You'll want to:
- learn some history,
- see natural beauty,
- visit religious sites,
- view historical and modern architecture,
- sample the various kinds of foods and drinks,
- view or take part in cultural activities,
- and meet and talk to the local people.
What places should you cover? Well roughly:
- Coast (Clyde or Fyfe)
Try to spend as much of your time doing the above list of things with the minimum or time spent in a car and the maximum time spent on the activities.
Day 1: Drive ~2hrs to Vindolanda or Housesteads on Hadrian's Wall from Lake Windermere. Take sandwiches and spend the remainder of the morning and lunch time viewing the site and learning the history. Continue on to Jedburgh, stop at the Abbey ruins for an hour and a bit. Considering the borders roads are quite exhausting to drive, you might decide to overnight just shy of Edinburgh proper.
Alt Day 1: Go to Birdoswald on Hadrian's Wall which is closer to the M6 than other sites and takes less time to go round. Spend an hour there then return to the M6 and drive north the New Lanark, which is also a UNESCO World Heritage Site. Take lunch in Lanark town there then explore New Lanark museums and visitor centres. I suggest overnighting in Lanark giving you the evening to walk up to the Falls of Clyde.
Day 2: A short drive into Edinburgh. If you are arriving during the Festival, give yourself an extra day here. Otherwise visit the castle, walk down the mile to St Gyles, lunch at a restaurant such as The Witchery, go through princes gardens and do some shopping on princes street. If you are fit, walk up one of the hills such as Carlton hill or Salisbury crags. I seriously doubt you should try to get to Inverness on this night. You've scheduled allot of walking in Edinburgh and it is a long drive on the A9 up to Inverness. The A9 gets my vote as the most boring road in the highlands. It goes near many interesting places but never actually too them. The west coast route up the A82 is far superia. I suggest instead that you overnight in Edinburgh so as to enjoy the night life of this city. Perhaps take in a show at one of the theatres.
Day 3: Drive past Stirling and head into the highlands. Stirling castle is nice but if you've already done Edinburgh castle I wouldn't go to both. Wallace monument is best seen from a distance if you want photographs, rather than close up. Can't comment on Bannockburn as I've not visited the centre there. Either go for Loch Tay side (used in the movie "Loch Ness" as it is more beautiful) or the Trossachs. If you really wanted to push the driving you could get west to the A82, and north through Glen Coe, Fort William and along the road to the Isles and Skye but only do this if the weather promises to be good. Skye is famous for it climbing. If your not a climber, you'll enjoy one of the other islands better. Rather I'd suggest to spend your time in the Trossachs, Loch Tay and perhaps Glen Coe. Overnight somewhere in the central highlands. Calladar or Killin are nice villages as is Fort William.
Day 4: Drive round the west side of Loch Lomond, cross over the hills to Loch Fyne and lunch at the Loch Fyne Seafood Restaurant. Continue on to Inverary and visit the castle and gaol. The ancient ruins around Kilmartin provide another aspect to the tour. Loop round Oban (nice distillery) by tea time and return to Glasgow to overnight there.
Day 5: Take the PSS Waverly down the Clyde river to the islands and return same day. Stop off at Bute, Arran or Dunoon for a few hours. Or, drive down the Clyde coast and take the regular ferries (leaving your car at the harbour). These islands are much more accessible and every bit as interesting as Skye.
Alt Day 5: Rather than returning to Glasgow on day 4, take the ferry to Mull and overnight there. Visit Ioana on day 5 and maybe Tobermory (another distillery). Then head to Glasgow in the Evening.
Day 6: Spend the day in Glasgow museums, shopping, food and nightlife. In the evening, take the train from Glasgow to Edinburgh and connect with your sleeper down to London.