Is my moms mystery snail dead?

Besides sniffing and poking it, you can also check what position the snail is in. If the body is hanging out slightly, it could very well be dead. If it's completely closed up so that you can only see the shell and the operculum, but not the body, then it's still alive. It likely wouldn't be your mom's fault if it's dead though, Mystery Snails are often mistreated by pet stores - they're underfed, kept in the wrong conditions and can be exposed to certain fish medicine which harms them, so it isn't unusual to lose a few you buy from the store.

As for temperature, Bettas and Mystery Snails have different preferences. Bettas like the temperature to be in the high 70s/low 80s, while Mystery Snails like low/mid 70s. Somewhere around 76-78 should be a good meeting point for both though.

If you test the snail by annoying him (pushing on his trapdoor) and taking him out of the water to smell him, it will just start the playing dead thing all over again. This time it will be longer before he starts moving and eating, so you don't really want to disturb him. There are things we need to know and things that are pure curiosity. When I fell down a flight of stairs, the doctor said I probably broke my tailbone, did I want X-rays? She said it would be idle curiosity to know since the treatment for a bruised or a broken tailbone would be the same. So I never had the X-ray. The snail does not need the X-ray treatment either. You will know soon enough if you maintain patience. Be sure though that there is no current rocking him, since in such a spot, the snail will stay closed up much longer.

Proper temp: 78 degrees, no less than 76 degrees

Snails sometimes won't move for a few days and appear dead. Take the snail out of the tank and push on it's "door". If there's resistance than the snail is still alive. Also, dead snails smell horrible. I'm sure your snail is fine.

Yeah anywhere from 70F-80F is an perfectly fine temp for a betta tank with snails. The snail is probably just adjusting but definitely do the "check" if after a day or two he still hasn't moved. Also be sure you're supplementing with algae wafers every 2-3 days to ensure he doesn't starve.

your snail very well could be dead

this sit can help you with yoru betta

http://www.bettafish.com/

if you see it not moving look at it closely then you will see if it is alive or dead