You pat them with the sock, yes. Some varieties are only on the birds at night, and on roosts/in bedding during the day. Yes, it will put some dust in the air, and whether you use a permethrin powder or Sevin, you should really wear a mask.
But at two weeks I think picking and just erratic feather growth are probably the cause. Where would they get the mites at two weeks? Are they outdoors? Mites and lice are ordinarily carried in by wild birds -- they can get them from the birds just flying over and dropping some --- but at two weeks?
If they are picking, they can definitely include the stomach area. Those might be the easiest to pick, actually. Are they perhaps too warm? How big is the brooder, for how many chicks?