Yes, you can. But “fancy” as a verb is very British English. “Do you fancy a cuppa?” Some people might not understand and might only use “fancy” as an adjective.

It would be more natural to say 'Do you fancy a chat tonight?' A talk (noun) is generally a speech to a group. It can also be a conversation about something specific, or with a specific purpose. As you're using the word 'fancy', I'm assuming that you're suggesting a casual conversation.

Yes you can. It is a phrase normally used by the British, but in social Conversations you can use it both for American and British English Conversation

: