‘Dorm-mate’ is a great try, but wouldn’t be too natural because ‘dorms’ are usually too big. It’s not close enough to merit the term ‘mate’. We would just say something like ‘we live in the same dorm’ or ‘we’re both residents of Baker House’ (a hypothetical name of a dorm). There are the terms ‘suite-mate’ which could describe living in the same suite of a dorm and ‘floor-mate’ if you live on the same floor of a dorm, and the floors are small enough and have some independent character as a social unit. Adding ‘mate’ implies that you should be close enough to see each other regularly and know something about each other’s habits.

Colloquially, you can certainly say "dorm-mate". Hope this helps :)

It doesn't exist at least not at the moment.

It does not. In English, however, it is OK to manufacture compound words yourself if that is really the best way to communicate. I can't actually think of a short, easy way to say "someone living in the same dorm as me." "Dorm-mate" would work in writing, and would be understood. I personally would rework the sentence. Instead of saying "My dorm-mate, Lewis Robinson" I would say "Lewis Robinson, who lives in my dorm."

No it doesn't exist, but consider a moment: Room-mate is made up of two words: room and mate. In English you could put any word you like in front of mate, to mean someone you share that thing with: housemate, flatmate, etc. Unfortunately you said dormate, and the reason this feels wrong to me is that "dor" isn't a complete word.