1password works on all of those except Linux. I believe LastPass works on all.
> Do they automatically sync?
1password syncs to Dropbox.
> Can I be sure they won't be shut down and thereby leave me cut of from all my online services?
Yes, if it syncs to something like Dropbox that has a local copy.
> And what do I do when the password manager is, inevitably, broken into?
The same thing you have to currently do if you're using one password everywhere?