For fresh snow late July or August is your best bet.
Legally you only need snow chains for Koskiousco Park if you don't have a 4wd but practically if you're not used to driving in snowy conditions 4wd won't stop you from sliding out so you may as well take them. You won't need chains for the trailer because it hardly ever snows in Jindy.
You have to pay to drive into the park - no idea how much it is for a day because we buy 2 year passes.
Selwyn is cheaper than Perisher or Thredbo if you're not into skiing but really skiing is awesome so you should give it a go.
The only resort in NSW that allows tobogganing is Selwyn but if you drive past Thredbo to Dead Horse Gap you can toboggan.
Perisher has tubing which is fast and doesn't reqiure any skill other than sitting down.

For snow I just wear 3 layers - base layer is thermal, second layer is fleece third layer is waterproof. How thick those layers are depends on the weather on the day. Layers are better than huge bulky coats because it can get hotter than you expect on a bluebird day.
You need sunglasses and sunscreen because snow is reflective and there is a lot of UV up high.
You need warm, waterproof shoes (gumboot are not warm).