Are Yii plugins compatible with craft? Specifically, I'm looking at https://github.com/yiiext/shopping-cart-component. Can Yii extensions be made to work with Craft (or work out of the box)? 

I'm unfamiliar with both, so I'm having a hard time approaching adding a shopping cart to this project, and I think rolling my own would turn out to be a mistake
No.  Yii's extensions and Craft's plugins are different things, although it's not that difficult to port a Yii extension to a Craft plugin.