The tablets are your dechlorinator. You'll need to use one at every water change to remove the chlorine from the new water. Once you've used them all, you'll find it easier to use a liquid one. If you read the bottles in the shop look for one that says it detoxifies ammonia. These have something in them that turns the ammonia into a less toxic form - that's less toxic not non-toxic. But the effect only lasts for about 24 hours so they are a way to help the fish between water changes, these products should not be used instead of water changes.
I'm not sure about a heater for a giant wine glass shaped tank. How warm is the room where the tank is? Bettas should be kept at around 78 deg F (25 deg C). If the room is that warm, it'd be OK without a heater. But if the room gets colder overnight in winter you will need some sort of heater.
You can't use the usual stick shaped heater as you wouldn't be able to attach it to the curved sides. If it's like a wine glass, I assume it's on a stalk? So a heater mat that goes under the tank can't be used either.
I don't know which country you are in, but if it's the USA you could try something like this http
/www.amazon.com/Submersible-Betta-Bowl-Heater-7-5w/dp/B006OP6C5M/ref=sr_1_4?ie=UTF8&qid=1346613364&sr=8-4&keywords=heater+betta+bowl Look on google for things like that.
I wouldn't worry too much about putting the shop water in with the betta. As you only have the betta in there and he'd been living in that water before you took him home, it should be OK. It's when you are adding more fish to a tank that you have to be careful because the water the new fish come in might have bugs that would infect the fish already in your tank.