"I have a mother."
Translation:У меня есть мама.
We don't omit "есть" when the primary focus is on having something/someone. It is okay to drop it when having is implied and it is important to clarify what/whom exactly you have.
- У меня мяч, а у тебя кукла. - I have a ball, and you have a doll.
- У меня красная машина. - literally I have a red car, but the meaning is closer to My car is red.
Да. There is no prior information to suggest that the 'existence' can be ignored.
This is a response to 'Do you have a mother?'. The existence of the mother is stressed so it must also be stressed in the reply. If you didn't already use the word есть in the same sentence, it'd generally be best to use it.