Hostas flower early summer May-July depending on cultivar and your grow zone. The best time to divide is now or early spring while still in bud stage.

different species flower at different times. my earliest one starts July and then the most of mine flower in August. one is repeat flowering this week.
I 've divided mine late September before but usually do them in the spring just as they start to grow.
The flowers if left on will go on to develop black seeds. They will germinate but may not be exactly like the parent plant.

I suppose you can take seeds from them but you need the patience of Mother Theresa to get decent size plants from seeds and you're never sure what you'll end up with. The best way to propagate is division - instant gratification.

Yes, mine seem to flower July till September depending on the variety. I prefer to divide in the autumn so the roots get a chance to get a grip before the soil freezes. But spring is just as good and you don't have to cut through the foliage, only the buds, so its a bit easier. I don't have the patience to grow them from seed. Like Bathgate, I prefer the slice em up method!