paper becomes a countable noun when there is something written on it, like a newspaper. So "The teacher passed out test papers" is correct example: "give me paper to write on" referring to a material to use for writing, uncountable noun object "show me your English test papers from last year" referring to papers which contain information, countable noun object

Blank paper is not countable. But a test paper is different--each test paper is unique because each student wrote a different paper. In this instance, "paper" means something like "essay" or "test," not "flat sheet of fibrous plant material." A test paper is countable, because each one is unique. -- "Papers" is also countable if it is referring to multiple sheets of a complete document--such as when referring to legal documents. Maybe you have heard of the videogame "Papers Please," referring to immigration papers.

A grammarian could probably explain this better, but “pass out the test papers” is correct. Otherwise it sounds like there is a single sheet of paper shared with many. The only way I think your correction could be used is if “test paper” were a type of paper being shared with the group. Construction paper for example. You could say, “The teacher passed out construction paper.” It’s unlikely anyone would assume you meant a single sheet in this case.

Thanks

He/she handed out the books. For "test paper" you could use it meaning there is one test paper with many copies, or you could refer to the many (booklets, instances) of test paper, that is the test papers. So in your sentence you culd use either one. Grammar describes what we say and write. If you think of it as plural, use plural, if a single entity, use singular.