In theory yes dogs are only supposed to be off the lead in the back section of the park, (and so I always stay away from that area), but I've frequently come across dog poo in other areas which suggests that there are plenty of dog owners who are either letting their dogs off the lead in the non-designated areas, or letting their dogs poop whilst they are on the lead and not picking it up.
I haven't come across any aggressive dogs in the Horniman gardens personally, (although I have come across them elsewhere in Forest Hill, and as I also have a curious and fearless toddler I am quite wary of them - particularly as one of those bull mastiff types almost went for my daughter not long ago. Luckily it was on a lead, and I was standing next to her, or I dread to think what could have happened).
It does seem a difficult combination though - a family-friendly museum, and a dog-exercise area!! Especially when the Horniman Park isn't exactly massive.
And as for the Triangle playground being 'a perfectly good swing park' I wouldn't call it that! Its adequate, but could certainly do with a bit of a revamp!