They (hydranges) normally do not need pruning. But if some of the stems typically dry out for you due to winter issues, you can prune then all the way down or up to 3-6". In the northernmost states, you may want to wait until mid-to-late May to prune though.

As for division of hydrangeas, do a hydrangea forum search for key words like dividing, etc. You will see that a lot of people divide hydrangeas successfully using a spade/etc but, technically speaking, they (hydrangeas) are not meant to be cut in half. Meaning, success is 100% guaranteed. So if you want to propagate a shrub that has some meaning to you, perhaps first propagate it via cuttings before dividing, just in case the shrub does not make it for some odd/unusual/weird reason. The best time for this action is probably when the shrub is dormant as that will reduce transplant shock.

I have Annabelle, which like Invincibelle is a Hydrangea arborescens. I only prune mine if it has been winter damaged by snow raked off the roof or if the rodents have chewed off all the bark. The second year stems are woodier and so less likely to flop. There tend to be many more flowers in the years I don't prune, though they may be a tad smaller. I do deadhead, removing the old flower clusters before the new leaves come out.

My Annabelle suckers vigorously, so I need to remove suckers most years. I. Often just rudely dig them up with a spading fork and cut off with a sharp shovel or pruning clippers. If I know someone who is interested, I toss into a bag, pot, or cardboard box with some damp soil. They have all taken hold in their new homes just fine. I have no direct experience with Invincibelle, but I doubt if it is too different.