Since you can only take reactions when it is not your turn, can you dodge a counterstrike since it would have been triggered by you on your own turn? Are the rules intentionally set up that way or is it just suppose to mean you can not take a reaction to get on extra action during your turn to say dodge for a little extra movement?

I would say yes, if you still have a reaction left, I wouldn't allow a parry since the opponent bypasses your blade.

Rules As Written, I believe you cannot take a Reaction in, well, reaction, to an attack made with something like Counterattack. Likewise, that would probably cross over into Overwatch fire, since you are attacked during your own turn.

Rules As Intended, however, I think that's all a bit silly and would let you make a reaction (providing you had one left, the reaction was otherwise viable, etc). Besides, I love the idea of watching two master swordsmen throw a bucket-of-dice worth of attacks at each other in a flurry of thrusts, parries, counter-parries, and counter-parry-parries. It's really quite badass.

That being said, that is different to the actual rules and it's always worth noting what the original rule is before you start messing about with houseruling.