Filters and bubble stones/wands are two different things. Filters, well they filter the water and remove some of the bad stuff and then you remove the others when you do a water change. Bubble stones/wands just put air into the water and agitate the surface which you don't want for Betta's. Betta's are surface/air breathers and need to breath air so when the surface is disrupted from bubbles, it's harder for them to breathe so usually we don't suggest getting air stones. It's not necessary.
Filters are up to you, if you want one great! if not, that's totally fine too! In smaller tanks it might be more of a hassle though but if you say get a 2.5 or bigger, you could definitely use a smaller filter without a problem but for a 1 gallon I wouldn't suggest it. You can do water changes every three days if you need, not every other but that's great! You're keeping the water really healthy for your little one
And yes! you can use tap water as long as you use the conditioner with it! Make sure to condition the water before your put your boy in it or add more water to the tank, that way it removes the chlorine, chloramine, heavy metals and other stuff that might be lurking in there. But yeah, tap is totally safe! Gosh, if I couldn't use tap I would not be in this hobby lol