By definition, unarmed attacks are:

Striking for damage with punches, kicks, and head butts is much like attacking with a melee weapon...

Which do not necessarily need a hand.

So, assuming somebody wields a greatsword, can he just use unarmed strikes as his off-hand attack option?

To push it one step further, can he TWF with, let's say, his left leg and right leg (for some tactical moves, like tripping an opponent multiple times) while holding a greatsword?

(I see an armor spike can do that, maybe so do unarmed strikes.)