No. When you select payed software like an arcade game and select to delete it, a message will pop up stating you can download it again for free from that profile of the marketplace while it's still up there.
Yeah, mine was broken too. Got a new one. Seems to be running good. Still, the Xbox 360 was a rushed console and therefor has problems. I will most likely sell mine and buy the new version when it launches. I bet the kinks are mostly worked out with that one.