Has anyone come across a software package that allows you to build animations in a similar way to how you can in Flash (i.e. using techniques such as tweens & bones & easings, etc) and then have the result exported as a sprite sheet?
You can use Flash, a 3D Tool or even Photoshops animation feature. Then render the frames to images and combine them to a sprite-sheet.
While those techniques work well for vector graphics, they don't translate well to pixel art. Sprites don't tend to handle rotation or scaling without distortion, so hand editing is essential.
Two products worth looking into for spriting are Cosmigo's ProMotion and GraphicsGale. Onion skinning and other features are available. But if you want clean pixel art, automation isn't going to do the job.
edit: Okay, my brain is tired. A cup of coffee later and I realize what you're looking for. You could try out SynFig, which looks like it might be adaptable to your needs :)
ToonBoom studio is a flash like animation tool and can export sprite sheets.
I've been using Texture Packer to create sprites for my iPhone game. It's similar to Zwoptex but has a free version. I'm porting a game originally developed in Flash, so I'm simply exporting each frame of the MovieClip to png and then importing those images in Texture Packer.