What are the options for creating/importing animations for a 2D game? One method is to use a sprite sheet and do frame-by-frame, but is there better/alternative ways for smoother animation? How about doing the animations in Flash and then importing into Unity?
All depends on what you need. There are several tools for puppet (type) rigs, including the built in animator. If you want bones and IK there are some tools like Spine, and a Puppet2D.
Spine: http://esotericsoftware.com/
Puppet2d: Puppet2D | Animation Tools | Unity Asset Store
And of course you can use Flash, but that can be a little tricky for puppet style animation, but definitely can be done.