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Hey I'm no expert on coldwater fish but I actually am running a tank with no heater at 71ish degrees as well. I have celestial pearl danios and they do well in cool water, and I'm sure they would appreciate the low filtration current.
White Cloud Minnows or Danios would be ok in a 10gallon. As far as i know from the list you wrote Neon tetra are tropical and need a heater. Platy can do well at 72 if your room temperature is warmest