Yes, mostly you'll see snow in Harbin, ten days would be too long for Harbin, you can check China Tours(https://www.travelchinaguide.com/tour/default.htm) for itineraries for nearby cities like Changchun, Shenyang and Tianjin. BTW, I think mostly your flight from Singapore will land at Beijing, if so, why not spend several days there and then heading to Harbin? That will make you a 5-7 days Beijing visit and then a 2-3 days Harbin tour. I think 2-3 days is totally enough for Harbin visit.