My cat is in heat and screaming. Is she in pain or just desperate?

This is normal behavior for a female cat in heat - Zinnie isn't in pain, just highly motivated to mate :)

Having her mated will temporarily fix the problem if she gets pregnant, but then you'll have to deal with all the possible medical issues of pregnancy and delivering kittens, and she will go into heat again almost immediately after giving birth. Persians and other flat-faced breeds are more prone to problems with having their kittens when they go into labor, so you should be prepared for the fact that she may need an emergency c-section when it's time to give birth.

If you don't want her to have kittens, a better option would be to have her spayed - this will keep her from ever going into heat again, and will also ensure that she can't become pregnant. If you do want to have one or more litters of kittens from her, I would recommend having her spayed as soon as you are finished breeding her for good - otherwise, she will continue to constantly go into heat and have the same issues she's having now.