To apply for Health card, no SIN card is required. You only need certain kind of document to prove you're living there --e.g. an envelop with your name and address on it.
You can write the driver's test any time you like. After that, what you need to do is just wait for the mail.
The health card, SIN card, and driver's license will be sent to you by mail. So at least there should be one family member stay in Canada for receiving it.