to write in / to fill in / to enter

You may (write/answer) either in pencil or in ink.
"on the paper" is correct but redundant.
This is normally the second sentence. The first will be something like, "Please answer the following questions in the blanks provided."

How do I feel? Here's how they parse:
(a) I'm supposed to write on the inside of the paper. How do I do that?
(b) I'm supposed to cover the entire surface of the paper with ink or pencil.
(c) I'm supposed to somehow go insde the paper. Where's the door?

(a) was a little more natural in that it required the least modification.

Usually you would find a brief instruction (you can use either pen or pencil) The person who wrote the form assumes that the test taker will understand that the instruction is meant for the form and applies to writing answers.

Thank you very much, you two!

Could you please answer one more question?
I've found this account about the term "to write in" in LDOCE:

2 write something ↔ in
= to write a piece of information in the space provided for it on a form or document

Judging from this, is it that you cannot use "to write in" in this meaning without an object ("something")??

To "write in" usually means an addition. In elections there are "write in" candidates whose names do not appear on the ballot and need to be "written in" by the voter. A person may sign a contract but "write in" an additional clause or condition.

To put something on a form would more often be referred to as "filling in".