First and foremost, does this component redirect for first time login?
Yes, absolutely.
I don't see that mentioned anywhere in the documentation.
I frankly don't really need the module for redirecting after the initial login, just for the initial login which CB handles differently. Do you support this and if so, how would I accomplish this?
Oversight on our part. I'll add it to the doc. All you do is put the relative URL:
index.php?option=com_loginredirectcb
in the CB configuration space for first time login. I have tested this and it works fine.