I'd say this is a situation where you can use both, as the usage of "gendered" language is changing. Maybe it's a bit special when talking about dogs, since we have the word "Hundefreund/Hundefreundin" which means dog lover. For this one, you have to use the female form: "Sie ist keine Hundefreundin". The expression "kein Freund von" is rather used with ideas or abstract concepts, such as: "Ich bin kein Freund von dieser Idee", which in my opinion is ok for both male and female speakers. And I think this phrase "kein Freund von" expresses a stronger rejection than the English phrase "not into something". As far as I understand it, when you say: "I am not into X" means that you don't care about X. So you are neither enthusiastic nor do you hate it. When you say "Ich bin kein Freund von X" in German, it means that you have a negative opinion on X.

Grammatically it would be 'keine Freundin', but in spoken German (at least in Austria) this (kein Freund von) is used by female and male speakers alike. I do not really recall people using 'keine Freundin" in this phrase, but it might be a regional thing.

Yes, for a woman it's the former, "keine Freundin".