The two school which seem to follow the British system are Tanglin Trust and Chatsworth. Tanglin will have a waiting list, Chatsworth may not. Chatsworth, Overseas Family School(OFS), and I believe United World College (UWC) follow the IB program. UWC have waiting lists. OFS never does. My Britiish neighbour's 3 daughters attended Chatsworth and were quite happy there until they moved onto India.
Yes, you will need to contact the school directly. Check the web. Each school should have their secretaries on during the summer months and most likely someone from admissions would be there to answer your questions.
You may be able to walk your daughter to school, this will depend on where you find accommodation. Some parents drive their children to school. Most schools, if not all, provide school buses which cost extra, but will pick up/drop off your child from the condo
guard house. Being a former teacher in Singapore, my students came all three ways: walking, driving, and by bus.
Most school have after school activities which may include your dancing requirements. I'm sure there are dance schools in Singapore, I just can't help with that one.
It is all a bit daunting: moving countries, finding accommodation and making sure your children are happy and settled. Remember they adapt to change a lot better and faster than we do! Good luck!