Starting the engine for short periods is a BAD idea. Unless the engine is THOROUGHLY warmed up for a PERIOD of time, you are creating nasty acids in the crankcase. A 30 minute warm up would be the bare minimum. I personally believe that you would be much better served by thoroughly warming up the engine, then changing the oil before storage. Then in the spring warm it up THOROUGHLY and then change it again.
Using inexpensive, albeit SJ rated oil for the winter storage would be okay, then put your favorite premium brand in when you change it in the spring.
Good luck,