For three gallons, with a filter you can try a couple shrimps.. but would be advisable to have live plants in there for them to have food and smaller cover as a lot of bettas will like to use them, or parts of them, as a snack if there isn't dense plant coverage on the bottom. Java fern, Anubias are easy, low maintenance plants. You wouldn't want to add in cories in anything under 10 gallons.. as the bio load would be too large, and cories do better in schools of 3+.
For shrimp care, I would wait for OFL to come back, as she will know more about their proper care.. I don't keep shrimp as I know they wouldn't last with any of mine.
Breeder so I have too many to count and too many named to list here!