Smaller tires will make the speedometer and odometer read incorrectly, reduce the handling of the car, result in a harsher ride, can adversely effect the ABS brakes, can result in a tire that doesn't have safe load carrying capacity and in short, be dangerous. Avoid any auto center that would put you in danger. For a 2013 Corolla, there are no smaller tires listed for it than a P195/65R-15. If they told you smaller tires were a listed option, they lied to you.
TireRack.com offers very good tires. I've been in the tire business for over 25 years and if I were in your shoes I would go for the Bridgestone Blizzak WS70. These tires offer the latest, most advanced technology for winter traction and do a great job. TireRack.com has them for $94 each and with a front wheel drive vehicle like your Corolla, you'll need four winter tires. Below is the link to the Bridgestone Blizzak at TireRack.com -
http://www.tirerack.com/tires/tires...autoYear=2013&autoModel=Corolla&autoModClar=L
Here is the link for this tire at Bridgestone's website -
http://www.bridgestonetire.com/tire/blizzak-ws70
When you purchase a set of Bridgestone tires you are also eligible to enter their Super Bowl Sweepstakes contest ->
http://www.bridgestonetire.com/#super-bowl-sweepstakes
If you would prefer not to have your tires shipped to you and installed, but to purchase locally and have them installed, here is the link to the Bridgestone store locator -
http://www.bridgestonetire.com/dealer
They even have a credit card you can get where they offer 6-months interest free financing, which is a great deal that I took advantage of. Below is the link -
https://www.cfna.com/wps/portal/bf
When the snow is coming down and everyone else is slipping and sliding, you'll love your Bridgestone Blizzak's! After your drive your Blizzak equipped car in a snowstorm, get back with us here and let us know what you think.
I hope this has been of help to you.