Have reviewed your install and I can't see anything within CB to allow this. The registrations have to be coming from some sort of API call as they don't have CB fields populated properly (approval and confirmation) and have no plan. They won't be able to login, but they'll still exist at its worst.

I think they maybe coming through Ads Manager as it seams to have the capability of registering users. Another possibility is AcyMailing. You've both these extensions to allow full public usage and it's possible they're creating a user so they can post ads or subscribe to newsletters.

You also appear to have no Captcha usage anywhere, this is a very bad idea for public forms. You should add CB Captcha for CBs registration process then enable, somehow, captcha for your adsmanager usage to avoid spam posting.

Also note you allow free registrations without a subscription within CBSubs > Settings > Global so a user registering without a plan is actually a perfectly acceptable usage on your site. As you have a free lifetime plan I suggest turning off free registrations without a subscription.

Please PM the username of a user this recently happened to so I can compare it to a normal user.

It sounds like they are cancelling their basket or their basket is expiring from non-payment. This can happen if they haven't paid in time before basket expires. You can adjust how long a basket is kept within CBSubs > Settings > Display > Baskets then adjusting "Payment basket timeouts" as needed. For example if it's set to 24hrs and if 24hrs is up then the basket is deleted. If you require confirmation before payment then perhaps they're waiting to long to confirm? This doesn't explain fields being empty though.

How must I do to cleanly uninstall and re install cb and cbsubs without loose settings?

Just install over top of it self. Don't uninstall. Not sure what a reinstall would do for you though unless files were corrupt or modified.