This is not possible by default in Joomla, although users of lower usergroups can only assign users to groups up to the level of their group.
Furthermore, Joomla's default user management occurs in the backend. I am not sure you would want to give backend access to so many users. You should look for a front-end solution.
Actually, it would be better to describe further your whole use-case and what you are trying to build and its purpose. Will that be a community, a membership site, something else...?
For what you are describing, I think there is one 3rd party component, that does this in the front-end, which I think is what you actually need and should do. The extensions is called DM Frontend User Manager, but it is a commercial one. I have no experience using this extension though.
If the extension does not fit your requirements, you may look for other similar extensions in JED. But depending on your actual final purpose, you might also look for communities, memberships and similar extensions. Otherwise, you will need to build something exactly for your needs (custom component, or a custom db app with a tool like fabrik, cobalt etc). This will take much efforts and if you can't do this at your own, you will have to pay someone expert.
Regarding the issues for the backend login for your custom usergroups I can't say much at the moment... You should review again all the actions you took when creating and assigning users/usergroups/permissions and make sure you have done everything correctly.
Have a look here: Menu Module not appearing/working with ACL, and let us know how this goes...