Do people I randomly run into keep living after I've left them?

All NPCs in the game follow a schedule, regardless of whether the player character is near their vicinity.

So they are technically still 'going on about their lives', even if you leave them alone.

However, some game locations will reset or respawn after some time (depending on their schedule and 'flag'). This will reset all NPCs that are based on that location.

From the UESP article, "Respawning":

Most locations in the game are respawning. This happens, depending on the location, at scheduled times. When a game location respawns, its enemies and loot are reset. This allows revisiting previously visited locations, which will have their contents reset.

In conclusion, I believe NPCs will not despawn until something causes them to die, or the location they're based in resets or respawns.