Hello from the central belt here in Scotland near the Forth, (waves).
I am sure you are checking out properties for sale in the free compilation of available listings, ESPC. I don't have a specific school to recommend art-wise though I'd ask the art school if there are particular feeder schools. Obviously if a particular school is oversubscribed there's a pretty broad selection in this region.
I don't surf but I expect you'll be looking at Dunbar for surfing, 27miles east of the City. Are you committed to Edinburgh itself or will you also include North Berwick and Musselburgh?
Up here the date of a child's birthday in connection with commencing school hinges on 1st March unlike English schools where 1st September is the key date.
So the older pupils in a class at my DD's state school will celebrate birthdays in Spring ie March and April. The younger pupils will be those born in December, January, and February.
In state schools pupils take the bulk of their Highers in S5 when they are 16. I am sure your DD is already preparing a portfolio. The Times will have listed the top performing schools. Did you know the head teacher at high school up here is often known as the Rector?
The autumn term begins in late August and summer terms ends around the end of June.
Apologies if you knew all this.
When looking at Edinburgh, do take a look at West Lothian, which is cheaper than the City, and served by good road and rail links, or if you are keen on surfing, East Lothian.