Tetras need bigger groups than 3, a group of 8 minimum is best for them
DON'T put any goldfish of any sort in a 10 gal, they won't grow much and you'll end up stunting them. Also, constant upgrades to coincide with the growth of the fish is expensive and leaves you with a lot of empty tank you may not have space for. With a large growing species like goldfish, it honestly is much cheaper for you ( in the long run ) , and safer for the fish to get a large tank to begin with ( a pair of fancies like moors needs around 55 gals )
A better idea for your unheated 10 gallon is to simply get a heater from ebay ( they're cheaper on there) as this will increase the types of fish you can choose from for a tank that size.
With a heater in both tanks you could go for some small rasboros, such as Galaxy rasboros, or chilli rasboros, or perhaps some tiny ember tetras or Normans lampeye killifish, you could try male endlers which are very small bright and active, or a group of female bettas, or even honey gouramis
For bottom dwellers, pygmy or Hasborous cories are good, or perhaps a bulldog plec.
Or if you just wanted a single showpiece fish, then either a single male betta or a dwarf gourami.
You could also divide the tank with a mesh divider to accomodate two male bettas, each in his own 5 gallon section.
Shrimp you could have quite a lot of in both tanks, as they produce such low waste it won't affect your water quality. Apple snails you have to be more careful wih as they do produce some noticable waste due to size. It's reccomended for 2 gallons per apple snail.
Considering the amount of choice you have with a heated tank of that size,compared to an unheated one , getting a heater is really not that much of an expense or inconvenience.