I think its best to ice a cake the day after its baked, as it tends to be less 'crumby'. Am I right in saying this?
Yes and no, really depends on the texture of the cake, if you can be bothered doing a crumb coat, if you want to bother with something under the fondant, etc.

I've frozen cakes plenty of times and never had an issue, moist cakes thaw out better though. If you're worried about dryness fill it with something moist.

Personally at this time of year if you're using butter cream I would keep it in the fridge, I've never had issues with that either.