I think the Jelly Club has closed.
It's really not that cold here, last winter not a flake of snow and maybe one day where there was a hint of frost. That was it. I think somewhere like North Yorkshire would be much colder, always seems to be a shit tonne of snow there. No way am I buying the DCs winter boots this year.
We go the the zoo every Christmas Day, have lunch in one of the cafes (you don't need to book the normal cafe - you might be able to have a formal Christmas lunch in the Mansion House and for that you would have to book) and have our main meal in the evening. Usually just one of the cafes is open but when we started going to the zoo several years ago they were all shut so this is a more recent development. Zoo was shut last Christmas Day but that was due to high winds (my DM said she was delighted, we insist she comes too and she claims to hate it but I think she likes it really).
The Dome looks great from the outside but I'd take a picture and move on, it's packed at Christmas and not exceptional (wouldn't occur to me to go there any other time of year). Go to Prestonfield House or if you want somewhere closer, the Signet Library is supposed to be the best place to go (you'd need to book).
Castle is very expensive. We have annual passes for Historic Scotland who run several sites and that makes it ok so it might be work looking into HS passes if you are going to travel about a bit (I think they do an "explorer" pass). I don't find the castle very interesting (Stirling castle is better) so I like knowing I can nip up for an hour without feeling I have to stay to get my money's worth.
I like walking through the old town at Christmas time as its natural prettiness is enhanced by the Christmas lights whereas what you can get in the centre is a bit more generic.
What you absolutely must do above all else is get a cardamom bun from one of the Peter's Yard bakeries and go to the Chocolate Tree in Bruntsfield for hot chocolate - life enhancing.