Does Casper come to a screeching halt when your Activation Code expires?

I think you stop getting access to updates and support.

Nope. As @Caitlin_M mentions, you lose access to additional updates, and any support calls would likely be chargeable items. but the licenses you purchased are yours to use for as long as you want. Its only the support options that you're paying for yearly.
Although its possible to stop paying for updates/support and keep using the product as is, its not advisable for a variety of, hopefully obvious reasons, and to my knowledge, very few customers go this route unless they were planning on phasing it out in favor of something else.

Thanks! Oh believe me that last thing I want to do is run unsupported.