1) If I can find out what type of filter cartridges I need, is there a place to look for them online besides Amazon?
I wouldn't get a filter, it's useless unless you intend to cycle a tank and cycling something smaller than a 5g is hard and dangerous for your fish as the cycle will be unstable.
Plus it may create too much current. So that would be some money you save.
However if you need a filter for your 10G, I recommend something with low or adjustable flow (or perhaps you may have to baffle it with a sponge so your betta won't get bothered by it) and you can look in craigslist, petco, and any pet store you may have near you.
Ebay is also an option.
Hang on back filters (Hob filter) and fluvial sponge filters are the most recommended for low budget betta keepers
2) The tank still had all of the accessories in it when I pulled it out and had not been cleaned. It was just drained after the last fish died. Is it enough for me to just clean everything with hot water and a sponge? I'm terrified of using soap on anything for fear that I won't get it all rinsed out. But I don't know if the hot water will be enough to make sure there's nothing in there that will make the fish sick.
You can always use a bleach without detergent waterred down to clean it and rinse rinse rinse rinse rinse rinse... a lot.
If not a mixture with vinegar and hot water can do. Also rinsing a lot is needed.
Or if they make you quesy you can always dip them in boiling water, though if it is made of plastic or rubber I'd pass on this method.
3) I have noticed my Betta tends to stay towards the bottom of his tank lately and I'm sure that's due to the cold setting in here in Tennessee. We have had unusually warm temperatures through December so it's just now really starting to drop in the house. So the next questions is... Can someone recommend a small heater for the new tank? Small is the key but I definitely want it to be powerful enough for the 2.5 gallon.
Well a good heater for you is an adjustable heater around 10-25w. Hydor heaters are well recommended and a nice fit for your tank.
4) Without getting too off-topic, we also inherited a 10 gallon tank with Goldfish several months ago. Sort of a rescue from a friend who didn't realize what he was getting into. We have a Gold Snail and a Yellow Rabbit Snail in that tank. The Yellow Rabbit Snail has had 4 babies in a matter of two weeks! Since the Goldfish tend to pick on the snails, and these babies are teeny tiny, we removed them from the 10 gallon and placed them with the Betta. He doesn't seem to mind them at all. Nipped at them once or twice and then left them alone. They've been with him for nearly a month now and everyone seems to be fine. Just curious as to what others think of this combination though. They will eventually get too large to all stay in the one tank but it seems like a safe place for them now.
Well I do recommend a filter for a 10g (something that will not create too much current for your betta), it will save you trouble regarding water changes,and you can always look up guides in cycling here in this forum.
Read about what cycling is and how to do it here:
I didn't know much about the snails but reading about them from here: http://www.liveaquaria.com/product/p...02&pcatid=3102
it seems they produce a lot of waste and may grow to be rather big-- 3 inches.
Not to mention the temperatures are a bit incompatible with the betta as bettas enjoy temperatures around 75-82F.
And the snails like 64-75F.
And I think keeping the tank at 75F would strain the snails.
You could give away the snails and the goldfish and start a community tank with the betta, or move the betta to a smaller tank (though downgrading a betta isn't usually recommended) and choose between keeping the goldfish or the snails, because it is rather overstocked. And in the long run overstocked tanks do cause diseases :( I'm sorry if this isn't something pleasant to hear, I just wish to help