Nash Equilibrium and Dominant Strategy
You're on the right track here. You need to check every outcome for its potential to be a NE. You're correct in stating that outcomes (5,10) and (10,5) are NEs however you didn't identify that (5,5) is also an NE.

(5,5)
If player one deviates he receives a payout of 5. 
If player two deviates he receives a payout of 5.

Therefore no player has an incentive to unilaterally deviate and (5,5) is a Nash equilibrium.

You're also correct that (5,5) is the dominant strategy solution. Just keep in mind that the dominant strategy solution will always be a Nash equilibrium but not all Nash equilibria are dominant strategy solutions.