How to remove these spots from my car's rubber trimming?

Just use a rubber trim dressing product. If you carefully mask off the paint then you may even find tyre shine will hide those signs of surface oxidation.

Whatever dripped on it looks to be permanent if it hasn't washed-off by conventional means.  Perhaps a rubber/trim coating/protectorent will even it out.

You can try cleaning them up with some tire cleaner on a rag - but it may have permanently damaged the rubber.

I think that whatever protective coating used to be on the window has basically flaked off and left exposed metal or something. Either that, OR the car wasn't regularly washed, so there was some sort of contaminant that sat there for a long time and leeched into the material.

The fact it does not scratch off with your fingernail says it is IN THE RUBBER.  
. Go to the dealership & let the parts guys know(show them the picture, if the car is there, that is even better that they can see it.  It may be a defect in the rubber(how it got past quality control is unusual. Most dealerships have their own bodyshop.  So THEY MAY HAVE A SOLUTION. Go with that first as the Dealer will try to make it right.

Kind off looks like tree sap, try bug and tar remover.
Another good cleaner is lighter fluid for hand held lighters.