Dáme si předkrm nejdřív ?
Do you have a question about this sentence? If yes, please do ask the question. Otherwise it will be just removed.
Let´s have an appetizer first? Is it also correct?
No, because that's not a question, it's a suggestion. That would be "Dejme si nejdřív předkrm" or "Pojďme si dát..."
OK. Thank you.
"Are we first having an appetizer?" should be accepted.
What is wrong with 'Shall we have a starter first?'
Nothing's wrong. I've added it now together with a few more options.
Is this the waitress asking me that or me asking to my wife before the waitress arrive?
It's you asking your wife or friend. The waitress would ask you using "dáte si..."