Gatlin, you want a lot.
I will try to get you pointed a little in the right direction.
#2 - Based upon your questions, you should not be the server admin.
#4 - Moodle supports Paypal
#5 - I think the answer to this is "Yes", if your admin know what they are doing. I am not sure what you mean by "secure the lecture videos," however.
#1 - It depends on how you run courses. If the instructors of all 20 courses, with 20 students in each, say "start the quiz RIGHT NOW" all at the same time, this might then be 400 concurrent users for about 5 seconds (approximately).
#3 - Moodle offers a variety of already beautiful themes, many which can be modified to a certain extent. Beauty is "in the eye of the beholder". I actually think that functionality comes first, even though I also think my Moodle is quite beautiful (thanks to some great moodle developers.)
My videos are roughly 7MB/minute. A lot of the size issues that you reference depends upon how the videos are compressed. If you could achieve 7MB/minute, you would then need only approximately 168GB (by my rough calculations).
Your last (large) paragraph suggests that you should not tackle this endeavor on your own, yet. Consider continuing your education by learning a little more from the moodle docs and from others hear on moodle.org. Some of the terminology will start making more sense to you. (By the way, no offense intended.)