Trying to figure out how to have gravity in a platformer based game, where the map moves in the background simulating vertical movement. however, the current issue is gravity does not match up with the pace of the screen moving down, and the collision is even more buggy.

The first thing is to start counting positions and speeds relative to 'game world', and then consider that there is an invisible rectangle, the camera (or 'screen window') that tracks the main character. This kind of system works equally well for side-view platformer and overhead (top down) games.

Anyway, if the character falls through platforms when he's falling fast enough, you must reconsider how you're handling collision. Checking collision after a character movement is too late.