"Is it morning now?"
I did the same, and got the sentence completely wrong. Is word order that important?
My guess is that since old or known information comes first, and we know that it is now, but not if it is morning, the word morning comes after now. But this is just a guess.
"Сейчас утро?" means "Is it morning now?", but how can I affirm that? How I could say "Now is morning"? I just would have to change the intonation so the listener knows if I'm asking or affirming?
Yes, you have to intonate the phrase to know if you are just saying or you are asking.
I believe "сейчас есть утро" would mean "the morning exists now", or something like that. But I am not a native speaker and couldn't say for sure.
In Russian, you don't normally use есть where the verb "to be" would be used in English.
Because we are asking about the present moment not about something unexpected that's already happened