"Does he have a mother?"
Translation:У него есть мама?
No. It is even stressed in this question.
When you ask whether someone has something, you have to use "есть". У тебя есть ручка? Do you have a pen?
If you say "У тебя ручка?", stressing the word "ручка", you will actually ask if you have a pen - and not something else. As if all the people in the room had pencils and this person had a pen. Like "Is it a pen you have?"
Hope this makes sense. "У тебя мама?" sounds really weird for a Russian ear. I can't think of a context to say so.
Yes, it does sound weird.
When you omit "есть" in such sentences, you focus on the object you have, not on the fact of having it. Sort of "It is a mom that you have".
У девочки банан, а у мальчика яблоко. - The girl has a banana, and the boy has an apple.
У меня красная машина, а у тебя синяя. - I have a red car, and you have a blue one. It is often said so to tell "My car is red, and yours is blue".
In both examples we focus on the objects and their differences.