Your first question has nothing to do with free will. We humans take the simplest thing and try to sound so philosophical and smart. Free will is having the ability to make a choice. Nothing more nothing less.
Placebos work because the subjects choose to take them and they believe they are taking something that works. The result is not because of free will, it is a result of free will (choosing to take the placebo). You’re talking about the after effects of free will.
Placebo’s work because your will believes they will work. Placebo’s actually work due to a manifestation of free will.
They are directly related.
Follow the logic of what you stated, “The result is not because of free will, it is a result of free will (choosing to take the placebo). You’re talking about the after effects of free will.”