I have a 03 coupe and the driver door window is suppose to drop 1/4 in or 39mm when the door is open.

Recently I realize the drop would probably drop about 5mm just, just enough not to chip the door cover.

Took it to the dealer, they replaced the window motor but I still have the same problem.

If I push it down it goes down to where it is suppose to be (1/4) and goes up normally when closing the window.

What could cause the window not to go down the 1/4 in drop, but i can push it down without a problem?
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The window motor has nothing to do with that. It must be re-initalized (programmed). I think you can do it yourself so it's probably posted somewhere on here how to DIY. If that does not fix it, you'll probably have to replace a door module.

 

The dealer is smart, they know best! the motor does allow the window to drop because i was able to just tap it down with my hand

 

I am having the exact same problem and have been searching for a while on this, let me know if you find anything

 

The window regulator will have excessive play regardless. Your door switch will tell the module the door is open/close and that the motor needs to drop/raise the window. The module only knows how much to move the window up/down but doesn't exactly know "where" the window glass is (exact position), only how much to drop/raise the window when they close the door. that's why they reset it and teach the module the glass is all the way "up". Other than a window regulator or hardware being loose/worn, it's likely in a module reset or a whole new module.