Mylar is DuPont's brand of BoPET film. There are others such as Hostaphan, etc. BOPP is different altogether. Another alternative is PEN but I haven't found any as thin as what's typically used here.
Each driver should have a direct connection to the bias line of the cable and gets connected to the diaphragm. In 6 pin Stax used 2 lines for bias along the entire length of the cable, in 5 pin they just split the bias line at the Y-splitter.
Yes, one goes to the inner stator and one goes to the outer stator. AFAIK absolute polarity isn't audible, but they need to be in phase with each other (ie. you can't have one channel with the + on the inside and the other channel with - on the inside). I've read in the past that almost all Stax are wired out of absolute phase, and I think that even showed up in Tyll's measurements.
There are websites that can explain this better and more in depth, but basically you have a thin, electrically charged film suspended between two conductive plates. The audio signal is amplified to a high voltage and as it runs through the plates, the stator with opposite polarity to the film pulls the film towards it while the stator with the same polarity as the film pushes it away (opposites attract, think of magnets). The plates have holes in them so that when the film vibrates it can push air through the holes and out of the driver, making sound. The amount of force applied to the film by a stator is a function of the distance between the two, and the closer they are together the stronger the force (think gravity/magnetism). Since two stators are used, we are able to keep a linear force on the diaphragm at all times under normal operating conditions.