1) at 11th level a mobile fighter gets to full attack and move, but he loses the first full BAB attack to do this, if he is TWF does he also lose the off hand attack associated with that first FULL BAB? IF so, if he only has TWF without improved or greater, does that mean he can't get any TWF benefits from this class ability?
2) can a TWF take improved unarmed strike, and Attack with his on hand as normal in a TWF sequence, have a shield is his off hand and use a kick for his OFF hand attack?
1) at 11th level a mobile fighter gets to full attack and move, but he loses the first full BAB attack to do this, if he is TWF does he also lose the off hand attack associated with that first FULL BAB? IF so, if he only has TWF without improved or greater, does that mean he can't get any TWF benefits from this class ability?
You probably should lose both the main and off hand attacks, but by RAW, you don't. You only lose a single attack at your highest bonus.
2) can a TWF take improved unarmed strike, and Attack with his on hand as normal in a TWF sequence, have a shield is his off hand and use a kick for his OFF hand attack?
You can, but why bother? If you have a feat to spare for Improved Unarmed Strike, why not just take Improved Shield Bash and dual wield your weapon and shield?
because Improved unarmed strike is a prerequisite for style feats AND lets you do unarmed attack damage if t's ever needed.
Yes, but it also requires investment in an Amulet of Mighty Fists to retain viability. It's much easier to invest in a magic shield, especially for overcoming material based damage types (since you can actually just get a shield made of a special material).
doesn't really need to be an investment of feats tr amulet of might fists..... enchanted boot dagger will do the trick.
Plus at a certain point when Ac doesn't matter enough anymore, drop the shield and use a normal light weapon.