I've found that Hong Kong Homes - Hong Kong Property, Apartment for Rent and for Sale - Hong Kong Realty - tends to be a better site for browsing property online, but the site is also aimed at more expensive properties/ areas to some extent, although they do cover some lower priced properties and may help you identify areas or buildings that you are interested in. Still, the prices are pretty realistic, perhaps minus 7-15% following negotiations with the landlord.
There are very few serviced apartments that will accommodate a dog, and they often book up months in advance, so if you have not secured housing before moving with your pet, you will need to make accommodation arrangements for it as well. Also, online sites typically don't indicate which buildings/ landlords are pet friendly. Once you identify possible areas to live in, make sure to tell the estate agent(s) you are working with that you have a dog and only want to see apartments in buildings that allow dogs AND that are owned by landlords who will rent to pet owners (make sure they call the landlord and confirm that pets are okay before you bother viewing) - will save you a lot of time. We had estate agents showing us all sorts of places that were totally unsuitable for dogs initially and I still don't know what they were thinking - that we would get rid of our dog after moving it from another country to HK?!