Thank you all for your help. I took your recommendations in my earlier threads and took my mystery snail out of the 1 gallon fish bowl that my Betta is in and my Betta is looking and feeling much better (not going up for air as much and seems more curious of the on goings around him). As for the snail, it is lazying out in its own container (I will get a bowl for it next paycheck). For whatever reason, the snail seems to mess up the bowl fairly quickly, even with me actively using a turkey baster to get its poop out of the tank.
My biggest enjoyment is having crystal clear water to see my little man swimming around, instead of a greenish haze from the algae (snail food) that seems to make a huge mess since they both were making a meal out of it. I notice the snail's container doesn't get hazy as fast, but it is already looking dirty, go figure.
The things that I have learned from here are:
1. The cup trick works best for my Betta, as he doesn't freak out as much and it is far easier to catch him (man are they fast
2. One gallon bowls (or tanks) are not enough for both 1 snail and 1 Betta, unless you enjoy cleaning every 2 to 3 days a week
3. Snails poop a lot and I mean a lot
4. Get a heater for your Betta to keep temps between 74F to 80F.
5. Mariomo (aka Moss) Balls help remove toxins from the water and Bettas seem to like them
6. Keep the bio load low for tanks less than five gallons (e.g. 1 Betta). Mariomo Balls don't add to bio load as far as I understand it
7. This place is full of good advice