Conditioned water doesn't necessarily have to sit at all. Its best to let it sit for 24 hours but you don't have to.
The betta cubes sold at petsmart are way too small. You need at least a 2 gallon tank. This one
from petsmart is a good choice, you can also get a bigger tank if you wanted.
You can name your fish whatever you want! Just pick a name that you like and think will fit him.