Also, I don't understand the three death saving throw thing - I had thought the character would stand up on their own after 3 saving throws. But none of mine ever did - they'd just stay downed with a full blue bar - they wouldn't continue making potentially failed throws, but they wouldn't recover either. Is this working as intended?
Yes, it's working as intended. You're attempting to stabilize. This is how it works in D&D 5e. You make 3 successful death saving throws and your character is stabilized, meaning they won't die if left unattended. They still require healing to be brought back to consciousness though.