They are exceedingly hardy - it's the national tree of Finland! But I don't think it is its cold-tolerance you are interested in. Birch wood is quite physically weak, that might be what interests you.

You can prune them hard and they recover. I once had one that split into two trunks not far above the ground level, and I didn't want that. I cut one of the two trunks off, effectively removing nearly half the tree, and it just grew away fine, it even straighted up so you'd never know it had once been two-stemmed. Given they grow in very windy places, like northern Norway, and aren't structurally very strong, they clearly need to be able to recover from natural mechanical breakages.

I would wait until after its leaves have fallen off before topping it. If you top it in the growing season, it will weep copious quantities of sap. Of course some people deliberately top them in the growing season to collect the sap to make an alcoholic beverage, known as birch wine, from it.