How many red cherry shrimp can I have in a 3 gallon tank with a betta?

No more than 2. It also depends on the personality of your fish.

I've had bettas that'll eat any shrimp in sight, and I've had bettas that won't even look at them. If you have a more docile betta, it'll probably be fine. If your tank is heavily planted, it may increase the shrimp's chance of survival. Get some anubias, java moss, amazon swords, marimo moss, etc so they can hide.

Good luck!

Sorry to say probably none. The Betta is very likely to track them down & eat them in such close quarters. "If it can fit into the mouth of another aquarium animal, it probably will."

In summer this may not be a big deal if your Betta is in water ranging 78-80 degrees F, but you might want to start looking around for a sponge filter, a submersible heater (they sell small ones pre-set at 78F) and perhaps a 5-gallon aquarium and glass hood. Maybe even get a small gravel vacuum so you can clean the fecal dirt out prior to your weekly 50% water change with treated water of the same temperature. Do you have a birthday coming up? Keep an eye out for Pet Store sales.

With those accessories and that care, your Betta's immune system will be far stronger. It may live well into its second year.

When you have the larger aquarium, just for fun put a small mirror up opposite your Betta. It is very natural for it to dance and offer threat displays to his "rival". That is called flaring. For a short time, so long as he doesn't jump out of the container, that does no harm. And when his rival "swims away" he will feel the pride of victory. If you listen carefully you can hear him exclaim, ":I'm da man!" ;)

Hope he is a delight to keep!

You shouldn't have any shrimp with your Betta. He will think they are fish food.

You can have 10. They have a very low bioload. But it depends on the betta. It might eat them.