It is payware, but reasonably affordable ($10/10 users, usually).
I've seen companies squeeze some (probably contract-breaching) huge numbers of users out of the very small plans of Atlassian software, as well, usually via insane multiple-people-using-same-account editing conventions.
Nope. They're completely separate things.
The hosted one they bought, and does mercurial too.
The self hosted one is what used to be called stash, and is git only afaik.
EDIT: I meant "host", not "hose", but given some of my experience sysadminning Atlassian products, the sentence may still be correct.
This is incorrect. Bitbucket Cloud and Server are two completely separate codebases. On the other hand, GitHub Enterprise is basically a snapshot of the production GitHub application.
There's not much that can go wrong on a single instance with a local database. Have a failover in case the HW fails and you are good to go, and it's faster. The pricing is not great, but perhaps that's something GitLab can solve.
But being hip and being lean (even if it costs more) is more important I guess.