This time around, I have copied an already working set of doors, changed the name of everything so the two pairs of doors/buttons doesn't affect each other, and fixed the direction of the opening doors' movement accordingly to being rotated from their primary position.
But somehow the right side of the door (the one named "right" and who's func_door is not no draw) sticks to the left one following it's movement.
I have checked every flag, output/input, and any other Hammer phenomenon I can think of, but the right door sticks to the left one's movement even though I have disabled the right half's movement!

Can someone among the more experienced community here please take a small look and correct my wrongs once again, please?

remove the parent from the door you dont want to move.

your door_1_right_door move direction is 0 0 0, which makes it move to the right ( as you look at it ). Switching it to 0 180 0 will make it move in the correct direction. The left doors movement direction is 0 0 90 and should be 0 0 0

This seemed to solve the problem for me. Let me know how it works out for you.

Hammer's direction/movement system kind of confuses me. I thought the opposite of 0 0 90 would be 0 0 270? The button/door wiki certainly implies so...?

An easy way to tell which way the door will open is to center the door in the top (x/y) viewport. Then go to the move direction property. Click and drag the little white line in the black circle to the direction you want it to open. This should make the door open in the proper direction. Sometimes it's hard to drag the line exactly on 90 degree increments, so just type in the correct number if its a few degrees off.

And yes, the opposite of 0 0 90 is 0 0 270.... however, you want to be changing the middle number, not the right-most number. And if I recall, your right door was 0 0 0 and your left door was 0 0 90 so... you weren't even using 270