I wrote the first option. The answer was accepted and then the system showed me the second choice as correct. I think the second is wrong. My opinion:
1) У тебя старая кровать? - Is your bed old?
The question is about the condition of the bed;
2) У тебя есть старая кровать? - Do you have an old bed?
The question is about the existence of the old bed.
From what I understand есть is understood in the sentence without it. So effectively the two are the same... Also the genitive would make the focus is possession not the condition of the bed. Those are my ramblings, for what it's worth.
Yes, "есть" imlies in the both sentence above, but a choice with "есть" focuses on an existence of an object of conversation. The first one doesn't have this focus, so the question is about the condition of the bed. The second one has this focus, so the question focuses on the existence of the bed, and the conversation is about an old bed, so the second sentence has the meaning: "Do you have any old bed?"
I would say you're correct, "у тебя старая кровать?" and "у тебя есть старая кровать?" do not seem the same to me. Similar to "you have an old bed?" and "do you have an old bed?" in English.
"you have an old bed?" this sounds odd to me without the "do", since it is a question
Thank goodness there are word options! I almost thought it said "Are you an old box?" which makes absolutely no sense whatsoever.
I slipped up big time!