hmm, skate skiing with a dog is a tough one. Most reliable answer would be at Winona Forest, which is about 2 hours drive north. The club up there grooms trails for skiers, snowmobilers, snow bikers, and mushers. Plus that area usually gets a lot more snow than down here due to lake effect off of Lake Ontario. They also host races of various types so you might want to check their calendar first if you don't want to show up on a really busy day. http://www.winonaforest.com/
For a more local answer your best bet is going to be the grooming done by the snowmobile clubs which is hard to predict. Their groomers tend to just appear at seemingly random times. Connecticut Hill is usually the most reliable but word is a) still not much snow up there yet, and b) there are some ongoing logging projects that are causing some normally unplowed roads there to get plowed this winter.
There is also occasional snowmobile grooming done at Hammond Hill and Yellow Barn but I haven't heard of or seen them come through yet this year. Seems like we should have enough snow now for them to start though. When they do groom you'll see their tracks next to Hammond Hill Road as you drive in to the parking lot, and then they continue on down the unplowed road beyond there. If you start skating from there it's a short drop and then a long uphill climb to the top of the hill. When it's groomed it's possible to do a large loop over to Caanan Road and then back to the parking by coming back downhill on Y1. If you attempt this you'll want to double-check that Caanan Road is in fact groomed before dropping down to it on skate skis. I've occasionally seen one road groomed and the other not.
I should mention one other little detail. As good as the skiing can sometimes be on the groomed roads, and no matter how deep the snow, once the snowmobiles start racing up and down they always manage to brings rocks up to the surface. So I never take a pair of skis out on those roads that I can't live with adding some new scratches to.