The apple juice might have some viable wild yeast present, or it may be yeast from the air inoculated the juice. Wild yeast has a low alcohol tolerance which it probably why your cider has been weak. You'll get better, more consistent results by adding cultured yeast. Champagne yeast is fine choice, if you like your hard cider without any residual sweetness, or are prepared to sweeten it just before drinking. Some ale yeasts will leave a small amount of residual sugar, but it's hard to replicate the semi-sweet taste of commercial hard cider without pasteurization and force-carbonation.
I would suggest a couple changes to your procedure. This is by no means the optimum way of making cider, but it should cost you nothing beyond what the juice and yeast cost.
Ferment the cider in bulk. Split one gallon of juice into two gallon jugs. Plastic jugs are OK. Fermentation will produce a lot of foam, so you'll want the headspace in your fermenter. Cotton balls in the neck of the jugs will help keep wild organisms out, but airlocks would be better. You can add sugar if you want to increase the alcohol percentage (you're living in a dorm, so you probably want this). Apple juice should produce between 5% and 7% ABV. You can add sugar to increase this, but if you add too much the alcohol produced will poison the yeast, and they may stop working before fermentation is complete. A pound of table sugar in a gallon of juice will add roughly 5% ABV.
Add yeast to fermenters and leave them alone for at least two weeks. Try to keep the temperature constant between 65 and 70 F. Consistent fermentation temperatures make for a better tasting cider.
When fermentation activity has ceased (a hydrometer would really be helpful here, but if the cider tastes dry and has started to clarify, that's an indication that fermentation has stopped) add priming sugar and bottle.
Skip the pasteurization step, unless you plan on re-sweetening the cider and force carbonating it.
The priming sugar will ferment in the bottles, producing carbonation. It's important that you get the amount right, or your bottles might explode. Use an online calculator.
Plastic pop bottles will work fine, provided they are clean. You'll know that the cider has carbonated when the bottles are firm to the touch. Chilling the carbonated cider for 24 hours helps get more CO2 into solution.
The white powder you saw could have been the beginnings of a bacterial infection, which is a common outcome when relying on wild yeast for fermentation. If it smells and tastes OK, it's unlikely to cause you any harm. But I'm not a doctor, so take my advice with a grain of salt.