1) Do weapons with the "grapple" special property, still force you to take a -4 penalty on your free grapple attempt (after a successful critical hit) for not having two hands free?
2) If you have Weapon Focus (monk weapon with the grapple property), Ascetic Style, and Binding Throw, You should be able to make a trip attempt with your chosen monk weapon, and still use Ki Throw/Binding Throw as though you had made an unarmed trip attack. When you make the Swift Action grapple attempt for Binding Throw, would you take a -4 penalty for not having both hands free?
1) I think currently, yes.
I'd need to look up the stuff for your second question, but the answer to the first question is you still take the penalties.
I understand the rules aspect and the following is not a disagreement with what these folks have already stated, but my opinion. I do not agree with the rules; a weapon that is made to used in a grapple should absolutely not impose that -4 penalty. The Dan bong for example is a real weapon that is used to aide in grappling your opponent (hence the +2 bonus), not hamper you in the long run.
I agree with Faelyn, however, I think since the Garrote specifically states that it does not impose the -4 penalty in it's description then it would be considered normal for these weapons to suffer the penalty.
Description: In order for you to use a garrote, your opponent must be helpless or unaware of you. You must make a grapple check (though you avoid the –4 penalty for not having two hands free) to successfully begin garroting your opponent.
1) Yes, unless specified (Garrote)
2) Yes. Ascetic Style lets you Ki Throw with your weapon. Binding throw says you can initiate a grapple as a swift action after using Ki Throw but isn't related to using an umarmed strike or weapon attack, so it would be as if you initiated a grapple under a normal circumstance, -4 if you don't have 2 free hands, except you are doing it as a swift action.
However, if you critical with a grappling weapon and you have Ascetic Style I would have to say you do not take the penalty because the weapon offers a grapple check being treated as if it were an unarmed strike... which is unarmed, which is considered 2 free hands.