I supposed the five year requirement you referred to follows from U.S.C. § 1052(f):
The Director may accept as prima facie evidence that the mark has become distinctive, as used on or in connection with the applicant’s goods in commerce, proof of substantially exclusive and continuous use thereof as a mark by the applicant in commerce for the five years ...
The Person and the University do need to come to an agreement. Universities usually have clear guidelines and arrangements for intellectual property, eg.
In line with UK legislation, the University owns all intellectual
property or other materials developed by its employees, unless
explicitly stated otherwise.
Students are not employees of the ...
You can make a derivative work if:
the original is not under copyright,
you are the copyright owner,
you hold a licence from the copyright owner that says you can, or
your usage is fair use or fair dealing as applicable.
For your proposal, the image is copyright, you don’t own it, you don’t have a licence and what you propose is neither fair use nor fair ...
Words themselves are not protected by copyright. Curated lists of words, however, are (what's protected is the artful collection of words chosen for a purpose). Hence Hasbro owns the copyright in the list of playable words, though it is a matter for future possible litigation to see if the courts agree.
If you have permission from the copyright holder, ...
Yes, they have copyrights
If there's "no agreements and no structure", then every single line of code and text content that's created by someone else for that service is copyrighted, and they have exclusive rights on it so you (and your LLC) are not allowed to use that without their explicit permission.
If they don't care now, they might start caring after ...
Distinctiveness can be acquired anytime. Five years' time period only provides prima-facie evidence for acquired distinctiveness filing with USPTO for registration in the principal register. This evidence may be repudiated by the examining attorney. Five years' time frame provides an opportunity to prove acquired distinctiveness. Not a guarantee. Good luck!