$10-$12k?!
I have about 6-7 acres that I mow and a 1200' driveway in northern NH to plow during the winter...with that money you can very easily buy a great high quality ZT mower and a dedicated plow truck.
My Ferris IS700Z zeroturn was about $5500 and my particular plow truck was $9k but its a 1st Gen Dodge Ram w/Cummins...you can find a great Ford/Chevy/Dodge reg cab 250/2500 with plow and 100k-150k miles for $5000 all day long by me. The one thing Id recommend is dont buy something so beat up/rusty that it ends up being a dedicated yard-only plow truck...I had one of those and sitting around 8 months a year did more harm to it than anything. Something you can actually use locally in summer to pick stuff up, go to the dump, etc will serve you better. Heck even a 1-ton dump truck would serve you well and can be found in the $6k-$8k range!
Coming from someone who mows a lot and plows a lot - dont try to get one machine to do it all...its much easier to have dedicated equipment. A truck with a plow is the way to go in my opinion and nothing beats a ZT mower.
If you wanted to spend a little less on mower and a little more on truck I think a nice lawn tractor w/mower deck would do ok for mowing too.