They do not stack. You can have only benefit from one effective size increase at any given time. It was covered in an FAQ, I don't have a link to the specific one, but I'm sure someone will come along with the link soon enough.
If you increased your weapon's effective size with an Impact enchant or Lead Blades, that would stack with the Size increase from Enlarge person, because one is a size increase to your weapon, and one is a size increase to yourself. But Impact would not stack with Lead Blades, because they both affect the weapon.
Strongjaw wouldn't work on a monk because a monk's fists are considered light weapons, not natural attacks. The monk would need to have a bite attack or some kind of natural attack for Strongjaw to apply.
Correct, monks are (and have been since 3.5 or longer) counted as both natural and manufactured weapons. Base class rules in CRB.
Usually a monk’s unarmed strikes deal lethal damage, but he can choose to deal nonlethal damage instead with no penalty on his attack roll. He has the same choice to deal lethal or nonlethal damage while grappling.
A monk’s unarmed strike is treated as both a manufactured weapon and a natural weapon for the purpose of spells and effects that enhance or improve either manufactured weapons or natural weapons.
You're absolutely right, my mistake.