I actually ran from Deathknell to Red Cloud Mesa with three Undead characters already and I believe only the Well cleansing quests are Tauren only. That was on a private server, so it might be different in the Beta and Classic.
In WOTLK on retail you surely could (I remember running around gathering as much rep as I could) and I really suppose it was doable even in Vanilla. There are soem rare exceptions, for example from all the quest given by Mull Thunderhorn, only Poision Water is doable by every horde, all the other quests are race locked. I don't know if there are other exceptions.
BrHule Okay so if you look at the totem quest for Mulgore ( https://classic.wowhead.com/quest=1519/call-of-earth ), there is no mention of race, just like the Warlock quests that are confirmed working.
Now if you look at the Hunter quests (NE: https://classic.wowhead.com/quest=6721/the-hunters-path / Orc/Troll: https://classic.wowhead.com/quest=6068/the-hunters-path ) They very specifically list which races can do them.
Looks like you're in luck!
P.S. There are some class quests on private servers that are doable, but are not actually doable by both races going by the Classic Wowhead, such as the early Priest quests, so watch out all.