Race makes more difference for Priests than for any other class because of racial spells...but it still makes very little difference in PVE and not too much difference in PVP. Furthermore, the racial spells get revised often, so there's no point in planning around them.
As for your second question: Trolls and Undead *can* quest in the BE starting area, except for the very earliest quests which are BE only. (And you couldn't finish some of them anyway--one requires you to Mana Tap, for example). I'd personally recommend finishing your own starter area quests, then when you head out of the Valley of Trials/Tirisfal glades, take the zepplin/walk to the Undercity and take the transporter there to Silvermoon, then head to the Silvermoon ruins and quest onward from there.
I also think you might need to visit a priest trainer in your race capital at level 20 to get your own race spells.