I was wondering whether there was a pet breeeding limit (because I remember hearing that there was a 50 limit on males, and something like 20 on females?) and whether there was a way to extend this? Thanks.

You can extend the breeding points of your pet with Yellow Fish Eggs, a special item available in events, or the woodland altar on Explore.

Each pet has its own breeding limit, visible on the top left of its profile page. You have to have researched its genes (the main muscle gene) in order to see it though.