Regarding any pair of nouns in general, if you're married, just ask yourself how your parents refer to you and your partner. Compare that to how your partner's parents refer to the two of you.
In a bygone age it might have been standard practice for both sets of parents to name the man first, but today each parent names their own child first. And usually no-one even notices both naming systems being used concurrently at family gatherings and the like.
The basic point is people usually name the more important of any such pair first.
Regarding cats and dogs, apart from the common idiom that it might be raining them in that order, there's no significant preference either way for other contexts. As the chart shows, 'twas ever thus. Some people are cat people, some prefer dogs. They tend to name their preferred pet type first.