Of course you can always do RAM previews in fullscreen mode and snatch the graphics card out put via HDMI/ DVI, but that may limit your option to actually edit stuff while the output is needed elsewhere... For that extra hardware like a Blackmagic card that can act as a comp preview device would be advisable...
Sorry, but you're trying to make AE do something it wasn't designed to do. If you want to integrate motion graphics from AE, you will have to render the animation. If it has transparency, you'll have to render in a codec that supports alpha channels.
You'll also need a playback device to play (and possibly loop) your animations. Since these are live webcasts, I expect you may also want to include lower-thirds, titles and other things. If this is so, you'll need to get a character generator; most CG's these days have the capability of incorporating animations like those made in AE.
I've preped for a bunch of web casting and rendering the videos is always the best. If you are doing things like lower thirds many webcasting setups allow you to type in the text and lay it over a lower 1/3 bg.
One last thought. PPro would be much better at doing this than AE. You could load up a bunch of sequences and output them via a capture card to your screencasting setup.