Any source of ammonia works, the fish flakes are fine. Add some gravel from the betta tank, too, to speed things up (they'll be housing some beneficial bacteria).
What filter do you have? Yeah, you just layer the media (as for horizontally or vertically, I guess that would depend on the media you're using and the filter). Whatever you have to catch general debris (usually looks like a sponge or something) should go on the bottom. After that, you can put any other media like poly filter. Carbon and/or ceramic rings should be towards the top. I agree, the Carbon won't be beneficial here. Activated Carbon is great for saltwater tanks but almost useless for freshwater. You don't really need poly filter/filter floss. It can be used to polish the water, though. I would look into getting a sponge to catch larger debris before the water hits the rest of the media. I find it odd that you can't find the ceramic rings, do you have the AquaClear brand? They have their own version and you don't need an AquaClear filter to use it.
Platies prefer cool, fast moving water. Bettas are tropical and need warm water without strong currents (because of their long fins). You might be able to get away with keeping them together by finding some middle ground (maybe 75 degrees with moderate current and plants to baffle the flow) but I would consider other tank mates. Corydoras catfish are much more adaptable to temperature and are quite peaceful. You'd want a small species like the pygmy cories. Some of the tetras are too nippy for a betta but they'd be worth a look. Cherry barbs can also make fine tank mates provided the water isn't TOO warm (they wouldn't be happy in the 80's). Cherry shrimp and amano shrimp would work with a betta that isn't particularly predatory. Oto cats can live with bettas but need a mature, stable tank. Endler's livebearers (basically wild-type, hardier guppies) can be great tank mates provided you find healthy ones (they're used as feeders). I just wanted to give you some more options--the platies could
work but there are better fish for your set-up
You generally quarantine in a separate tank. But if you just want to quarantine the initial fish before adding the betta, wait at least 4-5 weeks. Most diseases show themselves by then and you'll know the normal behavior of your new fish, they'll be eating well, etc.
Oh, and consider getting some hardy live plants. They'll help a TON with cycling your tank and also give you something to fall back on if you add fish too early by accident (the plants can absorb ammonia, possibly saving your fish from poisoning). Java ferns, water wisteria, and anubias are all hardy, low-light species that will stay small enough for a 10-gallon. They require no extra care besides being in the water and having some light.