I assume somebody needs information ..several questions on the page, name, address, etc. Fill out the entire application . Fill in ONE answer.

WHY? “fill out” is an English phrasal verb that means write in all the answers. It is not a verb to be translated directly by other languages with “write / ecriver / escribir / schriven.” When talking about a list of questions, we say, “fill out this form” in English.

PS: You put your hand INTO a bag. (Preposition of motion, into onto). Afterwards your hand is IN the bag.

It would be very difficult to get inside the page to write. I would say it is impossible.

You put your hand in a bag–it’s got an inside. Paper does not have an inside.