Sorry I'm late jumping in here, but I have a few things I'd like to contribute.
I have had many bettas that eat snails, so this is not unheard of. That is a very healthy diet for them, but should not be their only diet. Snails are high in protein. Bettas are, by nature, insectivores. Mosquito larvae is one of the best foods you can offer a betta, along with live black worms, tubifex worms (if he'll eat them), brine shirmp, and other high protein foods. Blood worms are lower on the nutritional scale and freeze dried blood worms can heavily pollute the water quickly. I tend to refer to bloodworms as betta candy.
In regard to your shrimp question, you could breed ghost shrimp for puffer food, just as easily as cherry shrimp. The benefit of the ghost shrimp is that they get much larger, so they go further. That is a good food source for most puffer species, just shouldn't be the only food source. (nor should snails)
Where snails are concerned, the species of snails make some difference depending on the species of puffer you choose to keep. Freshwater dwarf puffers tend not to like trumpet snails, but ram's horn snails are at the top of their list. Snails are not all created equal, and size is not the only thing that matters. If possible, find a species of snail that can thrive in the conditions the puffer lives in. This makes it easier for adding a large number of snails at a time so a few have a chance at laying eggs in the main tank before they are eaten. Be forewarned, if feeding a puffer... even a freshwater dwarf puffer, keeping live foods on hand by breeding them yourself is quite challenging and will likely require more than one tank. Supply/demand are not always on the same level, and the shrimp/snail tank(s) should be started long in advance of bringing home the fish you intend to feed them to. I can talk you through an easy set up for ghost shrimp breeding if you like. I currently have a 10 gallon ghostie tank running, have bred them in many different sized tanks. The best success I had with keeping a good food population thriving was in 55 gallons. You will also want to learn to tell the difference in sex of the shrimp and try to feed primarily male shrimp until your breeding colony gets going.
If you want to go the way of cherry shrimp, you can start with only 2 if you have a healthy set up to put them in, but for food purposes I would start with a colony of 1 male to 4 - 6 females minimum. You'll need to be sure to get male and female both in your colony. 1 or 2 males will handle as many females as you offer them, and the more females you have the more fry you have to work with. I currently have a 20 gallon cherry shrimp set up going, and I started with 2 shrimp... 1 male, 1 female. By the end of the first 6 months I had about 100 shrimp. I sell them in groups of 50 - 100/month now, 2 yrs later... and my tank always has at least 300 - 400 shrimp at any given time. They repopulate by 50 - 100 about once/month on their own. This is also keeping in mind that I only sell juvenile shrimp, so there are always adult females breeding. If I were to take my current population and move them to a 55 gallon tank, I would expect it to double over the course of the first 6 months. If you are in need of cherry shrimp and plants to get that type of set up started, let me know. I can ship and I am willing to send plants with them enough to get you started in a 10 gallon tank (I'll throw the plants in for free unless you want more than that). I also have ghost shrimp & plants enough to offer in the same way. Ghosties prefer and do better with java moss, cherries do better and thrive with naja grass. 99 cents each for the shrimp, of either species, plus shipping. Let me know in pm if that interests you. I am very limited on ghostie fry right now because I stopped breeding for quantity, but I can spare a few to get you started.
I have done and still do what you are talking about doing, just a warning that its not an "easy" thing to accomplish to keep up with the food supply for your fish. I have ram's horn snails breeding heavily in 10 of my 14 tanks. I had 3 dwarf puffers in a 10 gallon tank, and mine were fussy, would only eat live snails. (thats not typical but it does happen) To keep those 3 fish fed it took 25 - 30 small to medium size snails each day. They depleted my 10 tanks worth of snails quickly and I was always trying to find more. My snail tanks range in size from 65 - 120 gallons, so these are not small tanks. If you do this you will want to make sure you have the timing just right so you always have snails big enough and ready for the next feeding, and takes about 6 months to a yr to accomplish. (this applies for the shrimp as well) Also, depending on the species of snails you're working with, with ram's horn snails you could raise snails and shrimp together in the same tanks to save on space and maintenance. If you get plants from me you are sure to get ram's horn snails as well. If you're patient they will populate your tank soon enough. If you want shrimp let me know and I can toss some larger ram's horn snails in for you too.
Best of luck to you!