Non human Hawke

You wouldn't really had to change much with a dwarven Hawke since you can make Leandra's disownment be due to marrying someone of no birth and wealth instead of because she brought more magic into the lineage.

An elven Hawke is a different matter because the Dragon Age universe has that rule that half-elven children always look human, so you'd have to change both Malcolm and Leandra into full-blooded elves, which means the Amell family couldn't be nobility in the Kirkwall. If it wasn't for that rule, there wouldn't be a problem since regardless of how Kirkwall feels about elves, it is Leandra who is listed in the will, Leandra who arranges and attends the meeting with the Viscount and it is Leandra who technically owns the mansion and its estate, not her elven children. By the time Leandra dies in Act 2, Hawke has accumulated enough importance in the city that you can justify them not being at risk of losing their home due to bigotry: you can even boil it down to something as simple as the fact that Hawke saved the Viscount's son from mercenaries so the Viscount feels personally indebted and repays it by ignoring/shooting down anyone trying to make a fuss about the elves in Hightown. By Act 3, it is a non-issue because Hawke has just saved the town from qunari and become the Champion: canonically, mage Hawke is openly mage-y, so elf Hawke should be just fine as well. 

Without the rule, you can also do things like Carver (and Bethany) looking human like their mother - maybe it leads to Carver denying he has an elf for a father on his stay-at-home path because it means he wouldn't be able to join he templars.

I always found it interesting to ponder what would change/need to be changed in Dragon Age 2 if Hawke could be a different race. It still wouldn't ever have been feasible because Bioware didn't even have time to make the game the way they intended, but it is still fun to imagine.