Get the cotton, it has great taste and can hold more juice, specifically organic cotton is a must have. Organic cotton doesnt have all the pesticides and other chemicals regular cotton has. All organic cotton has is the hydrogen peroxide as a die, but hydrogen peroxide (H2O2) is broken down to H2O (water) and O (oxygen) when heated, so its pretty much safe. And single coils are alot different than dual coils, its all personal preference so i would recommend that you try both, experimenting, is all rebuilding is, until you have whats right for you.
As for the battery goes, I would HIGHLY recommend getting a new battery, preferably the best battery on the market right now, the Sony 18650 2100mAh VTC4's, as for being the safest and hardest hitting batteries. They are able to go all the way to 0.2 ohm's considering it to be "Safe". The batteries you currently have are not considered "safe" for rebuilding. And thats the most important thing about rebuilding, safety. Not only are they to safe, im guessing those batteries can only run 10Amps, and with rebuilding, that is VERY low. With 10 Amp batteries i would only consider going to as low as .5-.6 but i wouldnt go that low with those batteries.
EDIT:
Also it seems like your more of a beginner, I would suggest reading everything from battery safety, to ohms law. And when you rebuild your coil make sure you have a multimeter or atomizer ohm reader. Without either of the one I wouldnt recommend building a coil especially if your a beginner. Do what you want but just know that without knowledge, materials, or common sense, you can be putting yourself in danger. You dont want a battery exploding in your face, ever.