Yes, if there is no direct flight to Shanghai, you can get to Hangzhou first. I think Sigapine do had flight to Shanghai directly. No snowing in Nov-Dec here. If you will get to Hangzhou first, you should take airport bus to Hangzhou Railway Station and then take bullet train to Shanghai. It is a bit difficult for you have to go with two toddlers. You should better go directly to Shanghai.

Yes you can take the airport bus operate by airasia, the bus goes directly to Shanghai city, the ticket counter is just right in front of the arrival gate which you come out from inside. RMB100 per person, the bus takes around 2.5 to 3.5 hours, because this is an airasia bus, they will make sure passenger of airasia flight is able to catch the bus, so don worry. This would be the easiest way to get to Shanghai from Hangzhou airport if you are taking airasia flight. Good luck!