Hi,
You don't necessarily "need" flash (I've made flv videos with AE that work fine, and have never even set eyes on flash), so that isn't the real problem here.
I would need to know more details about your problem. As I understand it, you have discovered the dimensions your video must be (350 * 392) and have resized your comp to match, so that is no longer the problem. You are now experiencing some kind of distortion in your video. I'm not sure what you mean by "figures" (people?) and by "half the size of the flash screen": are they stretched somehow, or is the video undistorted and just takes up a small space in the browser?
As I'm not sure what's going on, it's difficult to say what needs to be done. I assume you've changed the video dimensions - did you change them in the render queue somehow, or are you changing them in the comp settings (selecting the comp in the project panel and hitting cntl+k, altering the dimensions, selecting ok, then right clicking on the video in the composition panel, choosing transform from the context menu, and hitting "fit to comp width/height" to scale the video up/down to fit the new dimensions)? Also, what were the original dimensions?
Another possibility might (and I stress 'might', as I'm not an expert in how the flv format works) be that you are using composition settings with non-square pixels (the type of pixels that are used in tv). Try setting them to square pixels (the kind used in computer monitors). That may - or may not - help.
Perhaps you could also post your prof's code too?
Simon Bonner
youtube.com/simonsaysFX