You're worrying too much! Your DS will be thrilled with it, whatever happens!
Anyway, I would say, no less than about 1cm thick. It's so difficult to tell you exactly, as I've been doing it for years and want to say 'that thick' and show you with my finger and thumb! 
Personally, when I have draped the cake with icing, I would cut roughly round the edge, but not to close. Then, work it in, until it's neat. (If it all goes very wrong, a cheat I have used, is to finish it with a roll of icing all the way round. (like a long sausage all the way round). Makes a perfectly acceptable finish.
Use a rolling pin to lift it off the work surface. Treat it like pastry, I suppose.
Use either icing sugar or cornflour on your work surface and rolling pin, btw.