Nostalrius Begins - Quality wow vanilla realm (1.12)

I think there is a chain quest called "A New Plague" given by Apothecary Johaan that has you killing darkhounds and murlocs. Is this quest Undead only?

I rolled a Tauren and he doesn't give the quest. I was able to do the quest where you poison the Scarlet prisoner dude in the basement, though.

Correct -- there are a handful of quests in classic WoW that are race-restricted. "A New Plague" is one such chain. Another chain is cleansing the wells in Mulgore (Tauren only).

My Orc had the same problem. Being an experienced Undead player, I knew all of the quests and knew when some were missing. They just won't give them all to other races. I assume every starting area has the same problem. You'll just have to take what you can get, if you're doing it for the reputation as I'd assume. I always quest Undead areas for the reputation myself. May as well quest it while it gives XP too, as opposed to the area that gives rep you don't give a dick about.