"Professor Muhammed's weird room" would be:
غُرْفةُ الأُسْتاذِ محمدٍ الغريبُ
"ghurfa(tu) 2al-2ustaadh(i) muHammad(in) 2al-ghariib(u)" - Formal Standard.
"Weird" is the attributive adj. of "Room" so then the ending sounds for both should be in the similar case, ie. nominative. Whilst, Muhammad is بدل of "Professor" and بدل should follow مبدل منه.
1) "So the noun is definite because it's referring to a known person (Mohammed)?"
Yes, it is! To simplify the matter, we can say it like that :))
2) Yes, the غرفةُ أُستاذٍ would be "a professor's room", we don't know exactly which professor is and which room is. It can be Muhammad's, Ibrahim's, Ali's, and so on.
ShabillaField asks a question: why "The Professor Muhammad's room is weird" is wrong?"
Hope some native English speaker (or some mods who create the course) would like to answer this. Thanks in advance. :))
Update: I think tsuj1g1r1 has answered the similar question. Please refer to his comment.