"Give me one egg please."
Translation:Дайте мені, будь ласка, одне яйце.
Sorry? «Дайте» is the suggested option.
It's not infinitive, it's imperative. The infinitive of «да́йте» is «да́ти».
No, there is a difference. «Дай» is used to people you address with «ти». I.e. you would say this to close friends, people seen as peers, often family members (but might be different in different regions — better ask), children.
«Да́йте» is used with the people you address with «ви». I.e. for people you don't know well, strangers, and most other people. Also, it's used when you're addressing more than one person.
«Мені» is 'me, to me'. It can be used with both «дай» and «да́йте», but it's not really neccessary: when you ask for something with «дай(те)», it's usually obvious you want the thing give to you.