I remember seeing something about how the premade "Origin Characters" have their own unique quest lines. Do I have to start a new game as them in order to access these quests? Will I miss out on anything by playing as a "custom" character and just having these characters in my party instead?
Yes, you're able to do each Origin Character's story quest when they're companions. If you play as one instead, you get a few extra internal monologues and minor things (such as Tara showing up in camp if you play as Gale), but that's all.
The exception to this is the Dark Urge origin, who is never available as a companion.
You can find his corpse inside Orin's bedroom though.