Question: Does CB keep a running record of how many users a broadcaster has kicked out of their room?

Short answer, yes. Below is a screenshot of where the broadcaster can see their ban list. Please also note if a user continues to get kicked / banned from multiple broadcaster rooms, Chaturbate will and do place a ban on the accounts or shadow ban (site silence). A shadow ban is a silence within a room, if a user receives this from Chaturbate they will be silenced for every single room, so a user may believe they are being ignored. Even their tip notes will be silenced.

Hope this helps.

Phil
CBRules