I found the solution to this problem. It really is quite simple, just take some paper towel, fold it a few times and gently squeeze it under the mirror where the water runs down the door. I then finish drying the truck and by the time I am done I just take out the pieces of paper. As for the water collecting, I haven't noticed since right now I am in the dead of winter. Referring to the water going down the windshield, I always shammy my truck from head to toe whenever I wash it.

 

I was just fine NOT noticing the water in the little squares. But NOW(after reading you post) thats the first thing I noticed when I drove off this morning!!
Your right about the problem being annoying

 

I noticed this to. It's annoying. I had a 2000 F-150 XLT 4X4 with the same problem. I would dry the truck then take it for a spin around the block which would blow the water out of the mirrors. Then I would wipe the doors with a damp towel and dry.