ONE computer is WAY MORE than enough for this! Really! The server is going to be doing practically nothing to serve 10 video streams.
Again, you are the only one who is going to know what bandwidth you need. Experiment with codecs and bitrates and decide what is going to give you acceptable quality. Then multiply the bitrate by the number of simultaneous streams you want to be able to serve.
Running ANY kind of webserver - but in particular a video server - out of your home is a really bad idea, and you are going to find almost nobody here who is doing so.
I think you need to begin at the beginning and start learning the basics of web hosting and creating websites. Serving videos isn't much different than serving anything else - except that it will suck up much more bandwidth. ;)