File size is driven by a number of factors, including compression codec, image size, bit rate, etc.
Uncompressed AVI is going to produce the largest file size. Quicktime, compressed with some codec, is going to produce a smaller file. Part of the trade-off is quality.
It's up to you to statr up a small notebook, comparing the results of outputting footage using a variety of codecs and qualities.
As far as your first question goes, for most situations, I don't see a whole lot of difference between rendering and exporting. Here's what I think the main difference is, though: Rendering just uses the project settings. So, for example, if your project is a Quicktime with Sorenson 3 compression, then rendering will produce that kind of file (or at least sufficient preview files to view the clip in more-or-less real time). Exporting, on the other hand, usually allows you to pick the final form, whether that's another Quicktime, an AVI, an MPEG, or whatever.
I might be wrong. The difference between "render" and "export" isn't really so crucial to you, however, you have to start studying the effects of your various codecs.
Good luck!
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Edward
Like Lovecraft? Know Photoshop? Got time for the Unspeakable?