You can give the button two actions: Action 1: Animation > Play. Action 2: Video > Play. That gets you part way there.
I don't know any way to have the video stopping by itself to trigger hiding the video. But you could create separate animate & start video and hide video buttons for a particular video.
This is just a work-around but it might work for you.
1. If you have control over the flash movie - add some frames at the start of the animation.
Lets say you're using default frame rate of 24fps and once you click the button the animation flies in for 1 second - add 24 frames at the start of the flash video of a still image of the first frame.
Then use the previous suggestion and put 2 actoins on the button. This would give the appearance of the video being paused as it is flying in - then start playing.
Regarding problem 2 -
I'm not sure if this would work for your project but I would suggest putting the animations on different pages. On your master page place your navigation buttons (Previous and Next and even a 'home' if you want it) and any graphics/backgrounds etc. Then the user can click 'next' once they're done with the first animation, and move to the next slide. Home buttons are good too becase people can go back to the start and choose where they want to go to. I use this often for chapter selection.
Hope this helps.