They aren't our host, they are only our registrar. They have the cheapest prices to purchasing domains and owning so many it makes it a great place for domain purchases. As a host I wouldn't use them.
We've been with a few hosts as we've made the moves around. We've been very happy with www.Zenex5ive.com
which we will very likely be with for a very long time.
I didn't mention anything about the registrar stuff and them contacting them because it's pretty far out there. A host nor a registrar will take down a site for hosting flash games. Illegal content and bittorrents, sure, but I'm almost positive that they have bigger fish to worry about then flash game sites. Without expensive lawyer fees, it really comes down to the operator of the website to be willing to take the stuff down.
Our registrar is VERY unlikely to do anything. Our host would give us time to explain so they don't lose the $800+ a month we pay them.
I truly doubt anything will come from flash developers getting pissed off and not telling me what games to remove. If the flash game isn't from Fox Studios, Universal Studios, etc. (which are smart enough to prevent their games from running on domains they don't want) the host will likely take no action, as more of the Internet is on my side and my way of thinking on this matter than the flash developers.