Are you using Joomla cache or 3rd party cache? Caching HTML forms is bad practice. Forms often contain security tokens that change on every page load. When a form is cached the token won't change. This causes issues like what you're seeing. You need to somehow exclude CBs registration page from caching (same for profile edit and any other page where CB outputs a form). Caching is meant for static data. For example a Joomla article that is unlikely to change and unlikely to have any form of dynamic content. It serves no purpose for dynamic pages so is particularly useless for the entirety of CB. If it is used with CB it needs a very short cache time (like 5 minutes), but you'll still hit that issue every now and again.

I'm not aware of any, but you can check Joomla.org extension directory for cache extensions. As far as I am aware Joomla cache should only cache what you exclusively tell it to. By default that's Joomla content and modules and typically enabled via the Joomla menu item params. If your site is primarily dynamic there won't be much point in enabling caching though as data will just rapidly go out of date if the duration is more than say 5-10 minutes.

There's probably a way to shut off the caching on a page by page basis. If possible we'll implement this to our spoof check functions so this issue won't happen anymore, but I'm not completely sure if such a feature even exists in Joomla. I've added a bug ticket to look into it further.

forge.joomlapolis.com/issues/5031