I just love when duolingo offers more than an option of translation and even so, I get it wrong.
In my experience, "Davvero" is used to agree with someone.
Like: "It's a nice day today."
is this used when you really want to say : no need , thanks but you shouldn't have done
No. This is used when you can't believe what happened to someone and ask him/her why that is so.
I think it would be translated as "really" if it is in the masculine: «proprio». Some examples would be: «È proprio qui!» = "It is right here!" and «Lei è proprio intelligente.» = "She is really intelligent."
Is davvero used always as when someone os questioning something? Like, REALLY? like if someone says "prendiamo la tigre dallo zoo" you ask "davvero? ?"