First of all I will declare my bias, that I am English and spent the first 47 years of my life living in England, but have chosen to take early retirement from my busy job and move to Skye. So you can guess that I love Scotland and Skye especially.
So, on to your questions. I think it is quite difficult to advise you without knowing more about your interests. You have 7 nights in busy cities (Glasgow and Edinburgh) and a further 2 nights in Inverness (to my mind a working town without much to attract the visitor, apart from its proximity to Culloden). Glasgow and Edinburgh are very different from each other, Glasgow a more industrial, contemporary city with some great art galleries, and Edinburgh with centuries of history, the castle, the Royal Mile, the Botanic Gardens, The Royal Yacht Britannia, proximity to Rosslyn Chapel etc.
If you are more interested in the scenery of Scotland and maybe getting out and doing some hiking then you probably have too many nights in cities. If however, you are more into museums, galleries, 'the bright lights', pub culture etc, then you might have the balance right.
You also don't say when you are planning to come to Scotland. The weather, while never completely predictable, seems to me to be more settled (and a bit drier) in May, June and September, and in July and August the midges can be an issue in the Highlands (especially if there's no wind) . In rural areas the opening hours for castle etc may be much more restricted in the winter and spring. For example Dunvegan castle here on Skye is usually only open from Easter until mid October.
You mention castles, but it's worth noting that the Royal Family use Balmoral in the summer and visiting is pretty limited, with (so I am told) only one room in the castle open to the public. The castle is almost invisible from the roadside, to protect the RF's privacy.
And so to Skye. I have travelled extensively to most parts of Europe, North America and Australia & New Zealand. I felt that Skye was special from the moment we first came here for a short holiday. The scenery is wild, dramatic, sometimes bleak (it's snowing as I write this), and I adore the coastal landscapes which create an ever changing colour palette and some amazing sunsets.
Is it very busy? Well, that depends on when you come. During the school holidays (early July to end of August), yes it can feel busy at the major tourist spots, e.g. Fairy Pools, Old Man of Storr, Neist Point Lighthouse, Quiraing. BUT, there are many, many equally beautiful spots on Skye that many visitors don't bother to discover (probably because they haven't been featured in a newspaper article or guidebook!). You can get a sense of what might be worth discovering by having a look at the excellent walking/hiking website www.walkhighlands.co.uk You will see that there are many, many suggested trails and we can go out for a hike in the middle of summer and know that we won't see another soul if we avoid the places I mentioned above.
You mention tourist infrastructure. Yes it can be pushed to breaking point in those busy summer months. We have 11,000 permanent residents and the property taxes from those residents goes to Highland Council (based in Inverness) who then have to prioritise how to spend that money to maintain all the infrastructure we need (not just public toilets and car parks!) to sustain daily life. Accommodation is limited. There are no huge, chain hotels, but that to me is part of the charm of the island. If you search you will find wonderful, individual B&Bs, small hotels, or even 'glamping' (posh camping) pods. Yes, you will need to book ahead of time, maybe a few months in order to get a good choice of the best accommodation. (I'm running out of space so will post this and continue on a new post).