Third-party software can do this. With the appropriate authentication, an app can be notified any time you ban someone, and can then share that (or you could choose to share that) with other streamers. Those streamers can then empower the app to ban those same users, under whatever definitions you choose.

The Twitch API can tell you about cooldowns, so look into your bot's features (or ask the bot author, if you're not a programmer) to create that command.

Agreed. The information IS available somewhere, as it can be found in the CSV download from https://dashboard.twitch.tv/channel-analytics - having this available in Helix would be hugely helpful.