Both my wife and I are currently residing in Singapore, and we are sending our packet 4 materials with a relative to submit to the Kolkata VFS. Does our relative need an authorization letter from us, or would the appointment letter suffice?

Also, shall she retain the appointment letter after submitting the original documents as we would need the letter for the medical interview as well as for the interview at the Mumbai consulate?

I'm not sure if the relative can submit the paperwork, try sending an email or calling the Kolkata Consulate about it.

I know your wife needs the appointment letter for the interview. Usually they never take it anyway, at the VFS. Any doubt, your relative can make a copy of the appointment letter and give it at the VFS office, if they do let her submit the documents for your wife.

Good luck and let us know how it goes!