Can it careen around trying to grapple and fight other ships?
Yes. The only thing it can't do is fire at another ship.
Can it still be fired upon, and what happens if it suffers more hits?
Enemy ships can still fire at it. Nothing would happen if it took more hull hits, but rigging and marine hits would still count if those aren't already maxed out too.
Are the striking and sinking rolls one-time events or do they happen each turn when the conditions apply, or only when fired on again?
Ships that are vulnerable to sinking or striking have to check for those events at the end of every turn during the Ship Status Check phase (steps C and D).