Yes. If your base attack bonus is +11, you can make the +11 attack with one weapon, the +6 attack with a different weapon, and the +1 attack with a third weapon, assuming all three are available to make attacks with (unarmed strikes are always available, since you can kick people).
You only incur TWF penalties if you are getting extra attacks: FAQ
Pretty much the title.
as mentioned above, yes. But, since your weapon attack is probably better than your unarmed strike, why would you want to?