If you want tank mates beyond a snail or shrimp, you really should have a filter or else you're going to be doing A LOT of water changes, and most other fish IME aren't as forgiving as bettas.
Personally I'm not a big neon fan as they have been way inbreed giving them low immune systems and genetic disorders. I'm more of a ember tetra fan for betta tanks as they have the same water requirements, if you want a schooling fish. But if you want a colorful fish like the neons, I'd suggest endler's livebearers, but only get the males.
But moving on to your other questions:
1. Yes they need a filter, I really like my AquaClears and my Tetra Whispers. They do a fabulous job at cleaning my tanks
2. They can nip fins, but they only reason that they would do that is if they don't have a school of friends. Neons are better with 6+, but the more the merrier. But since you only have a 10g I would stick around the 6-7 range of them. (this also applies to the lovely ember tetras, and endlers don't need schools! and they're smaller so 7 would be a perfect number of them, possibly 8 if your betta is docile and you're over filtering.)
3. Neons, embers and neons don't really have a preference when it comes to plants. But they LOVE live plants. My endlers love eating the microorganisms on the leaves. My bettas love their broad leaf plants like java ferns and anubais. To really give you a better idea on plant types I'd need to know what kind of lights you have, if you have any special substrate and if you use any fertilizers.
4. Nope, no other changes needed (if you get the filter). But for the first month or two you're going to need to do 30-50% water changes on your tank because you'll be cycling. However, if you are able to steal some filter media/decorations/gravel from a friends or reliable LFS it'll help you out a lot! And about the food, flakes; pellets; frozen; and live food are all great for the neons, embers, or endlers
5. Yup, just some left over food is great for them. When I had ghost shrimp they would often swim to the top of the tank when I fed, and grabbed the flakes from the top!
No question is silly
It's better to ask, than to kill.