If you want a flowering Pawlonia in your garden you need a lot of space. Allow an area at least 30 feet across, I'd say. It has huge leaves, casts dense shade, so there isn't much you can underplant it with, so it takes up lots of space. If you want a flowering tree and haven't got that much space, then you can't realistically expect to have a flowering Pawlonia, there are plenty of other choices.

It won't flower at all if you prune it back, full stop. They flower on mature wood, and not at all (or not very much) when they are small. No doubt if it was already flowering size you could judiciously cut a few bits out. But in general if you prune it back it won't flower the next year, whatever time of year you prune it. The people who cut it back hard each spring grow them for the "jungley" leaves, not the flowers.

I used to live in a first-floor room with Pawlonia "just outside" my window, except that the trunk was at least 20 feet away from the building, because it has such a spreading crown. But this was the best spot to see the flowers, they aren't so visible from below. So people often recommend you plant them on the lower part of a slope, so that people walking past on the upper part of the slope can see the flowers.