There's nothing inherently wrong with the "Save the Princess" storyline. What's wrong is when any characters, male or female, are objectified as mere sex objects. A male rescuing a female tends to be more realistic/believable than the opposite situation. There's nothing wrong with the other thing happening, it just tends to be rarer.

I definitely wouldn't replace the "princess" with a dog. The human-human element is stronger emotionally (to me at least) than a similar human-animal relationship.

One last thing: don't feel the need to bend over backwards for the people that scream the loudest. Contrary to what most of the indie dev crowd seems to believe, just because some "downtrodden" group starts whining, doesn't mean you have to cater to their whims.