To hyphen or not: cat person-turned-dog person vs. cat person turned dog person
Solution 1:
Generally, you hyphenate words that are linked together when you want to make sure the reader knows it's a single subject. So when you hyphenate, just think if it makes sense as a single verb or noun taken out of context.
So the 'best' way to write is...
I'm a cat-person turned into a cat-and-dog-person.
Here's my reasoning:
a) I'm a cat-person-turned-cat-and-dog-person: there are just too many parts, and the 'turned' word nullifies the portion before it. For example, if you were describing a glass of water would you call it an "empty-then-full-glass"?
b) I'm a cat person-turned-cat and dog person. A "person-turned-cat" doesn't sound like a thing, so hyphenation linking those words makes no sense.
c) I'm a cat-person turned cat-and-dog person. "cat-person" works, but a "cat-and-dog" isn't really a "thing", unless you're a mad scientist.
Solution 2:
There is no need to use any hyphens at all.
These are not compound adjectives. I understand the desire to keep lexical ideas together, but the sentence's syntax performs that duty.
Sincerly, a rock star turned copy editor.