Re: why has facebook stopped working on my nokia lumia 635?
In short: Yes, you can. But you might need to re-link certain contacts from your People Hub.
Unlike games and chat apps, the content in your Facebook app is actually downloaded from your Facebook account from the Facebook servers - uninstalling and then reinstalling the Facebook app should not delete your Facebook account from the Facebook servers, and your account should still be accessible through any web browser even without the app installed.
However, if you uninstall your Facebook app and if you haven't linked your Microsoft account to your Facebook account, you may lose your Facebook contacts from your People Hub. As soon as you reinstall the Facebook app, however (again) you should see your Facebook contacts get synchronized back to your phone - though you may need to re-link contacts that have multiple accounts.