No, that would be «У мужчи́н есть хле́б». «Мужчи́ны» can be either nominative plural or genitive singular, but after «у» we use the genitive case, not nominative.
Мужчи́на is declined like feminine nouns (but it's used with masculine adjective and verb forms). Other words that behave like this include па́па 'dad', Ди́ма 'Dima'.
Enquiring about possession doesn't require genitive form, maybe you've misunderstood something?
Generally, the forms used in questions are the same as in statements. So, if you make «У мужчины есть хлеб» 'The man has bread' into a question, it will be «У мужчины есть хлеб?» 'Does a man have bread' (or «Есть ли у мужчины хлеб?», which means the same). Since the statement has nominative case, the question also has it.
If you wanted to transform a negative statement into a question, you'd use genitive. Negative statement is «У мужчины нет хлеба» 'The man doesn't have bread', the question is «У мужчины нет хлеба?» 'Doesn't the man have bread?' (or «Нет ли у мужчины хлеба?», which means the same).
No, genitive is a different way of indicating possession. «У Ада́ма есть соба́ка» means 'Adam has a dog', «соба́ка Ада́ма» means 'Adam's dog'.