So, assuming somebody wields a greatsword, can he just use unarmed strikes as his off-hand attack option?
Yes, you can do that. Not really a lot to say on the matter: nothing says you can’t. There are some other options for that which work similarly (e.g. the armor spikes that you mention).
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?
The “while holding a greatsword” is zero problem here. This works equally well with or without it.
There is, however, an argument to be had that you can’t do this, even fully unarmed. Namely, the two-weapon fighting rules require you to attack with a second weapon. Arguably, you don’t have multiple different “unarmed strike” weapons—no matter what body part you use, it’s still the same, singular, “unarmed strike” weapon. And after you’ve used that, any attack with any body part would be an attack with the same “unarmed strike” weapon, and thus not allowed by the two-weapon fighting rules.
This is, obviously, dumb, but it is arguably what they wrote. No one should actually implement that rule, even if you’re convinced it is what they wrote, because it’s dumb. The FAQ says you can do it (page 68), and not being able to even contradicts at least one actual class feature (the city brawler barbarian gets “the effects of the Two-Weapon Fighting feat when fighting unarmed,” which makes no sense if you can’t use two-weapon fighting with two unarmed strikes).
Anyway, if you’re allowed to do it without a greatsword, adding a greatsword doesn’t cause any further problems.