Just throw a tarp on there and make sure you turn it on atleast 3 times a week so the engine doesn't sleep on ya. You'll be fine, trust me. I live in Salt Lake City and I cover up my R6 every winter and it runs just fine in the summer.
Trouble shifting gears can mean you need a transmission flush. Change the transmission fluid, if that doesn't improve the shifting it may need a new transmission. Changing the transmission fluid is pretty inexpensive. Changing the transmission is super expensive.
The sputtering and dying could...