[Ryan Rhea] "What player dimensions are a good size for this project to keep bandwidth costs as low as possible?"
It depends. Sorry too many variables and too much subjectivity.
You'll probably get the best efficiency (quality relative to file size) using the x264 variant of H.264 encoder. The frame size may depend on the legibility of your content. Sometimes you can cheat by lowering the frame rate which would increase the number of bits per pixel per frame but the compromise is the loss of temporal quality (motion smoothness). You may need to give the potential customer a variety of sizes depending on their internet connection speed.
[Ryan Rhea] "What video file format and codec should we use that will cover Android, iOS and Win/Mac? Or is there even one, single format that can hit all these?"
H.264. It may need different delivery methods though. H.264 itself works in Flash, HTML5 (but not some browsers), Quicktime, Windows Media Player 12 (comes with Windows 7), iOS and Android (but you may need to use Baseline Profile).
If you have Windows users, not on Windows 7 (therefore using WMP 11 or earlier), don't have Quicktime installed, don't have or can't handle Flash H.264 well, have older browsers that don't support HTML5 H.264, then you may need to provide a WM9 version as well. This might be the case with someone on a Windows XP machine a few years old with an old version of FireFox for example.
[Ryan Rhea] "When rendering to the smaller file size (600-ish pixels) ...about how many megs per finished minute should I expect (so I can better calculate bandwidth needs)?
"
Do you mean 640x360 or 640x480 for example? It depends on codec and content. I've seen things down around 400kbps on up to around 1000kbps for these sizes and have some some variations above and below. kilobits per second times number of seconds is your file size in bits. Then divide by 8 for bytes. You can use any number of bit rate calculators to figure out the file size.
[Ryan Rhea] "Is Akamai necessary, or do you know of a service that is more cost effective with quality hosting that allows us to put the video content behind the paywall? We don't expect a crushing amount of traffic right off the bat (if we're very, very lucky maybe as many as 300 subscribers to start, who will be able to watch as much as they want while they have an active subscription)."
You may need to price hunt as well as feature hunt. This might be a good place to look for preliminary information.
http://www.vidcompare.com/
