I visited Bhurban, Nathiagali and Kalabagh in last week of December 2013 and the roads were clear. However, i took note of weather forecast so that there is no snowfall when I am on road. There was of coarse no greenery and everything was covered with snow. Temperature at night in nathiagali was something like -4 outside and you really have to cover up from head to toe :). But enjoyed a lot. I think up till now there is snowfall in nathia gali ?
Abottabad should also be no issue, though someone else could guide you about Mansehra and Muzaffarabad. Don't have much experience with both these areas.
Have you driven car yourself to these areas before ?
I have been to Nathia gali in City 2007 (petrol) and the drive was just fine, sorry don't have any experience of corrolla but I think Corolla might have more powerful engine then City 2013, also not sure if 2011 has ABS brakes or not which city 2013 should have.
Whatever car you take do make sure you have coolant in your radiator and not just simple water, cause water might freeze overnight.