I could have sworn I replied to this again already, I guess I forgot to hit post :)
A betta is the best option on that tank size. Yes, a heater is needed as Bettas are tropical fish.
I would skip the snail in that tank size. I say this because most snails reproduce like mad. Yet, a solo mystery snail if male doesnt, and if female will lay easy to see pink sacs above water line that you can just scrap away. However, a mystery snail is easily 2+ inches and has a heavier bio load than you would think.
I get away with a mystery snail, betta, frog, and 2 otos in my 5 gal, but I have a 10 gal filter on it, live plants, and do regular water changes...otherwise, this tank would be full of dead fish.
So with betta and a ghost shrimp you should be fine (with a heater). (FYI, even cool water fish can have their water up to 75F, and tropical is usually around 78F depending upon the species as some like it warmer in the 80-82F range like Discus and Blue Rams.)
I agree to start with the ghost shrimp housed with the betta, if it gets ate no big deal. You certainly dont want to put a $5 cherry shrimp in there...that could be a very expensive betta snack! lol
Some bettas can be very nasty with tankmates. I have always been lucky with all mine. I have kept male bettas (veil & crowntails) with african dwarf frogs, neon tetra, mystery snail, oto cat, rasboras, danios, cories...all with no issues. Of course you can usually tell ahead of time if the betta is a nasty one. Most thought my current Betta (no name yet, have had him for a few months just havent named him) would be nasty, as he flares at my husband and friends...flares (no gills though) at the frog and snail once in a while too...but he never flares at me and loves to watch me cook since his tank is in the kitchen (5 gal acrylic). Anyways, my point is, you can sometimes tell how they will be...but this usually isnt a tell for shrimp as they are more prey than frogs, otos, snails, and other small fish.
Anyways, good luck, and keep us posted.
Another alternative would be to keep one guppy with the shrimp and a heater in the tank. But guppies seem to do better with the theory of more is merrier (same for space). Just food for thought.