Intellectual Property Rights

The fact that you are providing free educational services does not determine whether you are committing copyright infringement. Just like a professor can't go copy a book of short stories and hand the packet out to his 400 students at a public university, you can't take someone's idea wholesale and give it away simply because it's a good learning experience for the students. Of course, as you are only using part of the video, you might have a good "fair use" argument depending on how much you are using, and in what context. My understanding of archive.org is that the majority of the content is user uploaded, and that various restrictions may apply -- I took a quick sample and found some videos with Creative Commons licenses, and others that had copyright notices. I would take the video you want to sample from to a copyright attorney and ask for a specific opinion as to infringement.

You are oversimplifying things. Certain uses may qualify as fair use under the US Copyright Law and there are limited educational exemptions, but further analysis is needed to know whether or not they apply. Also, that only relates to US law--will your video be available abroad? The repository undoubtedly is not the copyright owner. It may have the right to grant licenses for limited uses so that should be explored. If not, the best policy is to seek permission from the copyright owner.

Actually, archive.org is not an "open source repository." It's just a web site of an organization that tries to take periodic captures of the entire web. Nothing you find on archive.org is public domain simply because you found it there. So what this means is that the fact you got the video from archive.org doesn't change the conventional copyright infringement analysis. Is the video copyrighted? (highly likely). Is your use a "fair use"? (possibly, if your use is truly free and only for educational purposes)

Yes you are oversimplifying. The fact that you provide "free educational services" is irrelevant. Educational institutions must comply with copyright laws. Further, archive.org is not an "open source responsitory", but even if it was this would not solve your problem. "Open Source" is often a code word for people who are philophically opposed to any intellectual property laws, but in truth I have found that many so-called "open source" works actually violate copyrights of others. You cannot assume simply becase you see the name "open source" or "creative commons licenses" that you are free to use the work. Would the fair use doctrine protect you? Maybe but this is a complicated analysis and fair use does not go as far as most people think. Bottom line---you need to consult with IP counsel and go over the details of your use of this video. Otherwise, you run the risk of being sued for infringement.