i have a great idea for you missgreen16, I've been working with maya+flash for a long time and I have come acrross some prefered methods through my experience.

first I recommend you take those background animations in maya and render them using maya vector. this will easily give you that flash cartoony look without having to mess with maya toon shaders and all. then theres 2 image formats i have found to be most preferred when you render. you can either use swift3dimporter(swft) which will give you editable vector type formats in flash but the problem is that they take much more memory and will severly lag your scenes. So I recommend you simply do a png image sequence instead. it takes much less memory and the looks are relatively good and it has transparencies so that whatever isnt rendered in maya can be scene through in flash. then import it to flash, and drag from your library to a layer and you will notice a keyframe on each frame of your layer. then you can easily do your flash animations right on top or behind. let me know if you have any questions about this ive tried many different ways of doing this.
i sent some examples of how it worked out for me only in my case i mostly used maya for the foreground elements rather than backgrounds: