The path from Old Peak Road up to the Peak is great for dogs. Plenty of dog walkers everyday so it is quite safe. However if your dog likes to eat rubbish (the odd chicken bone, leaf, twig, etc) then you may want to keep an eye out on the first few walks. My beagle eats everything so I am now aware of where the "hot spots" are. For example, there's a ledge at the start of the pedestrian-only path where food is often left for the wild dogs/cats/birds - I have no idea why people would do that - but my dog often jumps up and gobble down what he can find. So I leash him for that section and then it's off leash again.
Once at the top of the Peak there is the morning trail for a nice walk (about 0.5-1 hour depending on speed) and halfway through there are a few options to veer off for a real hike. One of my favourites is the Pok Fu Lam Reservoir hike, it takes about an hour and it loops back up to the peak.
I doubt you will be able to find a large field for your dog to romp about but there is a nice open park at Pok Fu Lam called Cyberport Park which is a good alternative. The Hong Kong Dog Rescue's volunteers usually frequent the place to let the dogs offleash, but recently there have been cases of a dog poisoner (different from the one at Bowen Rd) putting a pufferfish into the water tubs. I still consider the park fairly safe as long as you can provide ample water for your dogs.
These are just the nearby options for you to consider. There are always plenty of hiking routes in the New Territories but I sometimes find it too troublesome to bring my dog all the way out there. HK island has plenty to offer, and who can forget Dragon's Back? Beautiful hike and the dogs love it, especially since the route ends at Tai Long Wan beach for them to cool off. Dogs are technically not allowed on the beach but during the cooler months where there are fewer people, I see responsible owners with their dogs at the corner of the beach (near the lighthouse, furthest from the cafe/restaurant).
Welcome to HK. I hope this helps!