Re schools: you're probably OK for now, but I'd suggest visiting as soon as you get here and registering with the ones you like where appropriate - some are first come first serve, others competitive, and the school results don't necessarily line up with that in the way you might expect!
St Georges is certainly worth looking at, thigh the last I heard about it personally was very negative, relating to the school achieving good results by encouraging weaker girls to leave, rather than by good teaching: don't know whether there's any truth in that though. I think it may be the only all girls option, if that's important to you.
Mary Erskine's-Stewart Melville's has a complicated together/apart system, mixed for early years and sixth form but single sex in between. I've heard both very positive and very negative opinions!
George Heriot's and George Watson's are often bracketed together. They are both much cheaper than many independent schools, with larger classes on average, but very well thought of. Watson's is huge, which can be a problem at primary level; Heriot's is smaller but has a very cramped campus. Good teaching at both I hear. Seems to be "what it says on the tin": the things I've heard people say they dislike have been the things that are obvious from that description.
Loretto, out at Musselburgh, attracts rave reviews from some. Cargilfield, at Cramond, ditto. Location is obviously a concern in both cases, but some parents feel it's worth the treck from central Edinburgh and there are school buses. Cargilfield is a prep school, going to age 13 - the only serious option if you want to leave your options open including your DD going somewhere else competitive for senior school, as obviously the places with their own senior schools aren't very into preparing for entry elsewhere. Loretto like the others mentioned goes all through. Of course there are pros and cons to both patterns.
Fettes currently takes children from age 7 I think, but I heard a rumour they were extending down. Very expensive!
I don't think any of these are competitive entry like London preps (or NY elementary schools) - maybe GH and GW are the ones with most serious entry procedures, as I have heard of people not getting in! Think Loretto and Cargilfield are both first come first served (but may well fill early). Others dunno.
You may be able to narrow your options significantly by deciding what's important to you (eg what it is at St Georges that appeals).