Does the iPhone 5C still work after being unsupported?
Yeah, it would still work.

It wouldn't be a bad phone for very simple tasks like the ones you have listed.

You could run apps that support iOS 9(some apps require a certain minimum iOS). This would be assessed on an app-by-app basis. You might get lucky with some, but you should research the requirements of the specific apps you'd like.

You could sync it with iTunes to listen to music and podcasts.

You could still make phone calls.

You can use iCloud for iMessage, Find My Friends/iPhone and syncing contacts etc.

Overall the phone is still very much usable, but it may be frustratingly slow if you're used to newer models and I'd try to get one with at least 16gb if not the 32gb memory version. The 8gb model would have maybe 4gb of usable storage.

As a side note - there is still some stores that sell these phones, although most of them would be pre-owned/refurbished. €189 for a used one in Ireland from a reputable chain of stores. They seem to hold their value reasonably well, which shows that they are definitely a viable option.