Should I drain my winterized pool after a large rainstorm?
Yes, you will need to drain, perhaps multiple times depending on the amount of rain/snow you get.  I bought a Wayne PC4 pump to remove excess water (Caveat: I made a mistake and used it with my saltwater pool, so it's rusting).

Your mesh cover manual will provide guidance on the height of your cover relative to the water level.  If you drain too much, the mesh cover can tear under heavy snow.  I have a Meyco and it says about 18".  But in my case, I keep the pool at about 13" below the level of the mesh covers, which gives me 5" of clearance on my skimmers.