I would like to know if it is possible to export the ready-made art from Adobe After Effects to Flash and then use the template generator to send to CasparCG or do I always have to do everything through Adobe Flash?
Yes and no. In that broad sence either be false. What you can do is export prerendered stuff, for instance as PNG sequences, and import them into MovieClips or the main timeline. What you can not is simply export a finished design with all text layer and be able to make them dynamic. So it always take a bit of handcrafted Flash work to make a AE design becoming a template.
Probably someday there will be a way to go from a .mogrt to a html template, but that is currently no more than an idea for one of the next versions of Caspar.