Yes leave them there. They will be back next year.

If you remember to give them a feed with to]mato fertiliser it will help them to build up the bulbs with new flowers for next year.

Leave the green leaves to die down naturally, so the bulb can absorb energy. Remove the flower heads and the swelling seed box for the same reason.

Yes they can. Best to deadhead the seed heads. Tulips are iffy, depends what kind you planted. Some come back, others may not. It's good to refresh the bed each year with a few new bulbs, in the fall.

Tulips are more likely to come again if your ground is on the dry side over the summer. The ones that do best here are in the poorest bed.