I think since it's your first betta you should probably try and find the most lively looking fish there is.. instead of Walmart try Petsmart, or a local fish store instead, they may have more lively bettas. I would take a small bright coloured object (pen cap) or a tiny mirror, and see how they respond. If he puffs up his gills that will usually mean he's in better shape I believe. If none of the fish look very good where you are, just ask an employee when a new shipment is coming in, and be sure to arrive that day since they're usually doing better. Look at him closely, make sure there's no white specks, fluff, anything that looks like it could be a disease.
I would recommend, after finding the liveliest fish that you;
1. Put his cup/bag in the tank for 10-15 minutes before letting him out so he can adjust to the water temperature in the tank. Dumping them in can give them shock.
2. Buy a thermometer, they need 76-83F, if it's too low that could explain why your second betta just stayed at the bottom, you should buy a small heater (researching different heaters is useful as many small ones are quite inefficient)
3. Don't overfeed the fish, don't listen to what the fish food says, they almost always suggest too much (think that they want you to use more product so you buy more sooner..)
Your first fish was clearly sick before you got him, the second one probably was too. I agree not to let your little boy help with the cleaning, you can let him feed the fish, but make sure you hand him the food pieces as kids tend to grab too much. Also teach him not to tap on the tank if he doesn't know already, that could stress the fish a lot.
I hope it works out for you soon, I hate when kids get upset over a pet dying it is really sad
Be sure to post back when you get your new guy with how he's doing