Train from London to Edinburgh
The times won't change much for August — normally at least 1 train/hour, every 30 minutes for a lot of the day. All trains are run by East Coast and book as soon as you can — their website will allow you to set up an alert to tell you when the tickets are released.
All direct daytime trains between London and Edinburgh are operated by EastCoast. There are some slight technical differences between the different types of train used on different services, but 99% of passenger won't notice and won't care. Service standards at weekend (provision of food mainly) is usually different from weekday services.
The train timetable changes in May. Most years there is not much difference, but this year the EastCoastMainline timetable will have more changes to the timetable than usual.
The exact "working timetable" is only published 12 weeks before travel date — sometimes as little as 10 weeks. This is to cater for when parts of the line may be shut for engineering — though hopefully the train companies will not be so utterly stupid as to timetable any major work for Edinburgh during festival season.
One of the (many) confusing things about First class travel on trains in the UK is that there is huge variation in what First class is. You get a bigger seat (and if you're over 6ft that is important — personally I find the standard class seats too cramped for journeys of more than two hours).
After the May timetable change, the EastCoast service will also include some food service within the price of the First Class ticket — but the exact details are not clear yet.
If you want the flexibility to hop off en route you do not have to buy separate tickets. An open ticket gives you that flexibility. However, if you know in advance exactly which service you want to travel on, then an advance ticket is (usually but not always) cheaper — often dramatically cheaper. The counterpoint is that an advance ticket has zero (repeat ZERO) flexibility, you cannot change your travel plans at all with an advance ticket.
You really do need to think about the trade off between flexibility and cheap fare and decide what's more important to you.
One thought about hop on hop off? What are you going to do with your luggage?
As mentioned earlier, Edinburgh in August is beyond busy. Have you got accomodation sorted already?