First off, I (suspect) that those bullet connectors are simply plugged into an isolation block.
The power being produced by the charge coils in the stator has to go "somewhere" or else it would cause the bobbins to overheat, possibly melt and damage the stator (happens and usually takes out the ignition portion as well). An isolation block uses the motor itself to "eat up" the power being produced.
When you install the regulator you simply "unplug" them from where they are now and plug the into the regulator - the regulator becomes the (new) termination point for those wires - tee'ing them off would result in little/no charge since electricity looks for the easiest path to ground, and going through circuitry and then into a battery is "the hard way". Jumping the gap on a sparkplug, in the ignition side, is the "really hard way"
Once you have the regulator installed, never (as in never ever) run the motor without a battery connected. No battery = nowhere for the power being generated to go = fried regulator which then causes a fried stator.
As far as "what gauge" wire.
You are looking at no more than 7 amps of power over a run of approximately 15 feet (4 meters is what 13'ish feet - we have used the metric system since I was in high school but I never really caught on

).
So based on that you "could" use an 12 gauge wire (based on the amps plus the distance, 12 gauge is recommended to prevent loss due to resistance - tables that do not consider voltage loss would say that 18 gauge "could handle" the 7 amps in a 15 foot run).
Since you want to hook it to your battery I would suggest that you go with a heavier wire just for the sake of durability (and what I mean is, the battery being somewhat remote from the charge source in your case, a heavier wire like say an 8 gauge, while "un-necessary", wouldn't easily snap off if was pulled, tugged, hooked by a piece of gear etc), and an 8 gauge (or 6 for that matter) wouldn't break the bank to get 15 feet of it. Using "more wire" than necessary, aside from cost, in this application, has no real downside.