Ask HN: What tools exist to do live video manipulation for streaming/recording?

I am considering doing online piano instruction, and am wondering how I can combine different video sources on the fly, and either broadcast or save the output. An example of what I am talking about would be https://www.youtube.com/watch?v=obNAgImLMSI, about 15 seconds in, where an overhead shot of the keyboard appears at the bottom of the screen. A different type of effect would be this: https://www.youtube.com/watch?v=stsBqdo9Svc, in which a 2nd keyboard appears showing which keys are being played as well as the note names (which can be difficult to otherwise determine in fast passages). I'm a good programmer (MS in CS), and don't mind getting my hands dirty in a bit of custom coding, but I don't even know where to start. (Don't worry, I'm competent to teach piano, too... I also have a BM in Piano Performance.) I specialize in teaching the integration of music reading and improvisation, and have had many requests to teach long distance, but I have never done so because I do not feel as though I have the proper tools to do it justice. I would love to find an Open Source solution that I could contribute back to, if possible. Does anyone have any ideas/recommendations?