What happens to cases that are dropped?

What is the name of your state (only U.S. law)? NY

Cases where charges were never filed and cases where charges were filed.

Are they filed away somewhere? Stored? Or just lost in a judicial abyss?

Also what happens to evidence from cases that are dropped?

Just out of curiosity, no specific situation or question.

That depends on the circumstances in which they were dropped. You may wish to Google "dismissed with prejudice" as a starting point for your research.