"This gentleman is my uncle."
Translation:Ten pan to mój wujek.
Then there would be no purpose to the lesson - it is important here that we address the man respectfully, so it's either "gentleman" or "mister", I suppose. We Russians, for example, too have borrowed the word "gentleman", but we use it either when we're talking about Englishmen or when we stress that this man has good manners, while the general word that means the same as "gentleman/mister" in English or "pan" in Polish and would be translated as such is "господин" (gospodin). So no, nothing really wrong about it.
No, your sentence is wrong. You should either say "Ten pan to mój wujek" (Nominative + "to" + Nominative) or preferably "Ten pan jest moim wujkiem" (Nominative + a form of "być" + Instrumental).
Your "mojego wujek" is a mix of cases that not only does not match the sentence, but even each other.
"wujem" is among the accepted answers. Technically, "wujek" is a diminutive of "wuj". But as such 'more formal' versions seem rather too formal nowadays, they aren't really used often. Besides, it sounds very close to a very popular swearword, so it's another reason to avoid it ;)
The word we teach is "wujek", Instrumental: "wujkiem". That's the main answer.
What was suggested to you, "wujem", is the Instrumental of "wuj". "wuj" is like the augmentative of "wujek". Kinda like "ojciec" to "tata". But it's not used often and may sound as if you dislike this uncle. It's rather formal. Also it sounds very close to one of the most basic swearwords.