How do I send animated GIFs in e-mail and know that they will be accepted and seen as animations in all e-mail applications?

You don't! Since, it's the e-mail client that is responsible for playing it and you have no control over what e-mail client people use or how they configure it.

That's the point. You can't control what e-mail client is used or how its configured therefore you can't ensure that its played.

That's the point. You can't control what e-mail client is used or how its configured therefore you can't ensure that its played.
I realized that you can improve the situation by putting the gif-animation on a site and providing a link to it. Not a complete solution but fairly passable.
My thinking was the result of the challenge of your very correct reply. Thanks ---and I'm not being ironic, it is this passing the question back to the urgent asker that produces the solution.:)