Is there always a pure Nash equilibrium in a resource selection game?
Yes, there is always a pure Nash equilibrium. See:

I Milchtaich (1996). Congestion games with player-specific payoff functions. 
Games and economic behavior 13 (1), 111-124.

You are interested in the special case of singleton congestion games with player-specific payoff functions.

And yes, they can be computed in polynomial time. See Corollary 7 in:

Heiner Ackermann, Heiko Röglin, Berthold Vöcking (2009). Pure Nash equilibria in player-specific and weighted congestion games. Theoretical Computer Science 410 (17), 1552-1563.