Hi,

  1. No, no need to be logged in to Gmail.
  2. If phone is off, app cannot be activated. However, it will activate once phone is on and SIM card is changed.
  3. GPS cannot be turned on remotely since Android 2.3.

For PIN or pattern lock it is hard to say. When your phone is stolen and not locked, thief can easily access your data before you send SMS or command from my.avast.
If its locked, then thief will maybe flash it. You can you Avast Anti-Theft for rooted device, which can survive flash.