Is one phrase more commonly used than the other? If so, what are the situations or relationships in which this would be appropriate?
I think the sentence with the word есть is more commonly used and that «У меня вопрос» puts the stress on the last word, вопрос. I imagine it would be used if the reciever (whom the speaker is talking to) is aware that the speaker has something, but doesn’t know what. Another context: У меня вопрос, а не ответ – I have a question, not an answer.
Is it ok if I write "у меня есть один ещë вопрос?" to say "I have one more question?".
I am wondering, because here they use it without есть, but in a different task, where they asked me to write "Do you have questions", I was marked as wrong when I used it without the есть.
Can we make it that it either both accepts both, or can someone explain to me why it differs between those sentences?
Есть = "exists" = "is" in the sense of "being" as opposed to "not-being". When the existence of something is at issue in the format beginning with У... you have to use есть when asking whether something exists or not.
In "Do you have questions?" the existence of questions is at issue, therefore you have to use есть. In an idiomatic way, the question is at its core, "Do questions exists [which you want to ask]?"
In "I have a question" the existence of the question is not at issue, thus есть is not necessary. You cannot "have" a question which does not exist; it has to exist before you can ask it - thus "exists" is not necessary because it it assumed.
That's not to say that у меня есть вопрос is wrong - I don't believe it is, just that it may be redundant.