Please find my answer inline for your query:
Ques: I want to get the list of friends from the linked Facebook user (user log in through FB and the account is linked with App42). Which API should I use for this?
Ans: You need to integrate Facebook SDK, please have a look at this link.
Ques: After I have the list of friends, which API should I use to send friends sign up request with my app/App42?
Ans: Once you have the list of those friends, you can send request on their Facebook wall using the Facebook SDK. Please have a look at above link and feel free to write us in case you need any help from our side.
We are happy to help you.