Yes, you can use whatever form builder extension you like or if you prefer to just create the form yourself using HTML that's also fine and have it submit to an auto action. Example auto action as follows.

Global
Triggers: None
Type: Registration
User: Self
Access: STAFF_ACL_HERE
Action
Username: [post_username]
Email: [post_email]
Groups: Registered
First Name: [post_firstname]

For STAFF_ACL_HERE be sure to select whatever usergroup the special staff user has assigned. This will ensure only they can use this auto action. Notice how the data is being populated from POST substitutions and the name of the substitution will just be whatever the input name is in your form. You can set the values for all kinds of fields from this auto action as well and will handle the entire user creation process for you.

You might be able to by creating a Auto Join field then try setting its value from the registration auto action otherwise it'll get a lot more complicated and require chaining auto actions together.

The field works with group ids. I've never tried doing this so it may not even work, but it's possible I could make adjustments so it can. Try using the |*| delimiter which is what we store array values with. So the following for example.

GROUP_ID_1|*|GROUP_ID_2|*|GROUP_ID_3