In answer to your questions:
Is it unprofessional/rude to communicate via Facebook?
No more so than using Slack or Skype for Business, at least for what you say they are using it for. Facebook is a communication tool, and they are using it as such. With that said, I would steer clear of talking about sensitive stuff on Facebook, which your colleagues seem to be doing. Even then, I would only do this because Facebook has a bad privacy reputation and, should it get leaked to the press that sensitive information is being communicated over Facebook, they may have a field day with it (there are a few laws that would also apply, but that is moot point for the purposes of your question). In terms of others eavesdropping on your conversations, Facebook is as bulletproof as Slack et al. If you are worried about your own privacy, just set up a burner account.
It may be that your signal problems are due to your carrier alone, and you would have these problems with any internet-based communications. Can you ask for a work phone?
Is it illegal?
Absolutely not and for the same reasons I've just outlined above. Checking your Facebook every now and again can hardly be considered work, and such an accusation would not fly in any court of law.
Am being I a sensitive baby about being expected to spend my off-the-clock time scouring Facebook (which I literally log onto about once every 2-3 months)
Yes. If you log in via phone, you should be automatically getting notifications whenever a post is made. Tapping that notification takes a few seconds.
Facebook is a SOCIAL network.
So is LinkedIn. What is your point?