Don't move it until it's dormant, which would be in late autumn onwards, (unless the ground is frozen).

I've never tried to split a Hydrangea, so I can't answer that, I'm afraid. However, it's very easy to take cuttings from them, if you want to increase the number in your garden.

I've never come across the concept of splitting a shrub or bush. Herbaceous perennials - yes, bushes - no.
You'd be better off taking cuttings as Spritz suggests.

Can't be split because its a shrub, as Beattie indicates.