Mini-split for cooling only (not heat) -- do I need the outdoor part?
Yes. you need both pieces. An air conditioner is a heat pump that moves heat from inside to outside. A dual mode system that can supply heat just runs the same process "in reverse." 

A heat pump draws heat from some source (often air) and transfers it into your house. An air conditioner is the same thing, but instead of drawing "free" heat from some thermal source, it draws heat from inside, and transfers it outside.

Your window A/C unit simply incorporates all of these pieces into one box, which necessarily hangs in your window so that they evaporator coil can be inside and the condenser coil outside. A "mini-split" system just separates these two, and is just an alternate installation style of what is commonly referred to as central air. The only difference is that instead of integrating into built-in ductwork (which might not be there), it instead hangs a unit on your wall. That unit incorporates the evaporator coil and fan, much like the inside half of your window AC. You still need the other half as well.

The image below is from Wikipedia