We just moved this past summer (Americans but moved from the UK)
We brought our medium sized dog - no quarantine when coming from the UK. We found most condos
would accepts dogs no problem - didn't look at HDB. Where we are, we don't have trouble walking the dog in our neighborhood. Lots of grass to sniff. We have had to kennel her twice in the early day when we were out of town and had no one to dog-sit. The kennel we used offered collection/delivery service. I know there is also a Pet Taxi you can book which we will use for vet appointments or something. Dogs are supposed to always be leashed here so I don't see much need to take her to the beach or something where she will still have to be on leash.
Our kids (Age 5 and 8) are also at Chatsworth. I chose it based on IB program, size, location and cost. We wanted to live centrally so we wouldn't need a car and Chatsworth is right off Orchard Road. I am very happy with it so far. I can only speak to the primary program though - not secondary which your eldest would be getting into next year I believe. It isn't nearly as big as SAS and there are almost certainly less extracurriculars than at the large schools, but what they have is plenty for my kids. Lots of kids here in Singapore go for private lessons anyway (piano, Mandarin, tennis, swimming, dance, etc).