Does moving a grappled foe through someone’s threatened area trigger an attack of opportunity?

If one has grappled a foe, and then succeeds on a check to maintain it, one of the options is:

You can move both yourself and your target up to half your speed. At the end of your movement, you can place your target in any square adjacent to you. If you attempt to place your foe in a hazardous location, such as in a wall of fire or over a pit, the target receives a free attempt to break your grapple with a +4 bonus.

The attack of opportunity (AoO) rules state that:

Moving out of a threatened square usually provokes attacks of opportunity from threatening opponents. There are two common methods of avoiding such an attack—the 5-foot step and the withdraw action.

  1. Does being moved out of a threatened square by someone else (in general, or while grappled) triggers an AoO?
  2. If it does, would that trigger the "if you attempt to to place your foe in a hazardous location" clause?