So Two Weapon Style says you use full defense and be able to attack. Does that mean you split your dice pool or is that 2 separate tests in a complex action?

As I understand it, no.

Normally when you use the Full Defense option when you're attacked, you sacrifice your next IP and get your full Reaction + whatever to evade an attack. With the Two Weapon Style you're using one of those weapons to defend with instead of attacking. Your off hand weapon suffers a -2 DP penalty, be it for defense or attack (unless you've got Ambidexterity or Off-Hand Training for that weapon).

Now since it says you're using the Full Defense option with one of those weapons, it seems to be that you're actually locked into performing a Full Parry with whichever weapon you choose to use, in order to get this benefit. The other two options, Dodge & Gymnastics Dodge, don't use weapons.

Note: this is an assumption that you as asking about 5th edition, if not, disregard this message

Not entirely, Full Defense is an interrupt action that can be taken anytime during yours or another person's IP. You reduce your total initiative count by the listed amount for Full Defense (10) when you choose to use that interrupt. From there you add your willpower to any relevant Defense tests you make during the round so for instance, if you are making a Parry with Full Defense you would roll your Reaction+Intuition+Willpower+(Melee Weapon) with a limit of your Physical Limit (if a skill is used, if no skill is used there is no limit added) for a Dodge it would be Reaction+Intuition+Willpower+Gymnastics with Physical as your limit (both of these options would cost a total of 15 initiative, should your initiative be lowered to the point that you can no longer take an additional action afterward, you lose any additional attacks)

Two Weapon Style is a Martial Arts maneuver out Arsenal, it gives you the ability to attack and defend with two weapons.

You get to defend with full parry (doubling your base skill dice) without having to sacrifice an action, although like SpellBinder said, you still suffer off-hand penalties unless you have a quality that negates it. You attack normally, on your initiative, and defend when you are attacked. The attack and defense rolls are separate.