Bettas can live without gravel or a filter, but for me instead of just bare bottom, I use marbles and shells, I do have a filter, and a light it helps keep the water warm, although it does get cold too my Aquarium is not as large as yours it's only a two gallon but it's perfect for a Betta and two snails, I also bought a plant the snails help keep it clean thats what they are there for, and if they can they will eat the betta's food too, I use "Complete Water Conditioner" to keep the water from becoming cloudy it helps, I clean out my tank once a week or two, if it starts to get too much algae thats were the algae eater comes in, which I do have plans to buy a small "Otociniclus catfis algae eater" to help with the algae and my Betta will be set,, He's happy.
and as for the Snail Floating my two snails are pretty active, today after taking a nap, I notice my betta touching the mystery snail t was floating, my Betta was checking it out I thought it died, but apparently after reading snails will do that at times, to collect air, it finally fell to the bottom and is active again. One person joke thinking it's rediculus to worry about snails but when you invest you do take an interest. I have a blonde snail too they are interesting peaceful creatures, and my betta Siren gets along with them and seems to take an interest in them too, they seem like a happy family. By the way Siren lets me know if he needs his water clean believe it or not he gets more active trying to get my attention more then usual. He's a prince on a pea.