Do I pop those plastic things out of the door and get new ones? I've given up trying to slide it on to those existing plastic things.

 

Sometimes (but rarely) you can get them out and reuse them.
I'd just go get some new ones.
This kit probably has some you can use:

 

That is funny. It looks impossible, even with new pins, since one of the mounting points on the back side of the carpet panel has a slot that points to the side in one direction, and the other two mounting points have slots that point in the opposite direction. It looks like a cruel joke on the part of the designer. How does a person slide the panel to one side and lock onto two of the pins, and then slide the other direction somehow and get it to click on the other pin without unlocking the first two pins?

 

You don't slide the panel.
You put the pins in the panel first then snap it in place.

 

Thanks I'll buy that kit.