With a normal game you'll be limited to playing races core to each alliance.

However if you own the Explorer's Pack you'll be able to play any race in any alliance, additionally owning the Imperial Edition gives you Imperials in any alliance.

There are no benefits or negatives to playing a race outside of their normal alliance, simply choose the race you want to play and the faction who you want to fight for.
There may be some odd instances where playing a Nord in DC might have NPC's you speak with insult Nords or have you fight other Nords but mechanically nothing changes.