We use WebRTC instead of skype, but yeah that sort of stuff is possible. When we revisit the mic situation while rolling out the teaching tools we can consider adding video support in as well. The primary problem with video is bandwidth, but as faster internet slowly gets rolled out that problem goes away 
I think video chat is more like blingbling feature than a usable at teaching. It’s nice to see who you chat with but it sounds something so far under the priority list that there would never be time to implement it.
Using skype or hangouts for audio is also not ideal. Both services require logins. Audio chat is best when it works easily with 1 click without any extra accounts and hassle.
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Forget Skype.
“As we look ahead to the future, we‘re focusing our efforts on bringing the latest and greatest to the most recent versions of Skype. As a result, we are going to retire older versions of Skype for Windows desktop (6.13 and below) as well as Skype for Mac (6.14 and below) over the next few months.”
Source: http://blogs.skype.com/2014/06/20/making-way-for-the-next-generation-of-skype-on-desktop/
As I understand this, it means that older OS won’t be able to run Skype.
I don’t think neither skype nor google hangouts are meant to be embedded into other applications anyways. Using WebRTC is way more reasonable. It is basically tailor made for the task.
How about instead of using skype how about using teamspeak you can have chat rooms no video though but i dont thinks its needed it costs a certain amount of money to keep it up but you could start a donation page on the website to keep it going.
Google Hangouts. You can display your screen during a video hangout. You can also interact with everyone invited to the stream. If you don’t have a webcam, you can still use the video chat for screen sharing and sound.
It’s the simplest way I have found to start streaming almost immediately. No need for it to be built in.
Skype works well for me and my nephew.
When you change to your browser window, the Skype image become small and you can easily drag it where it is convenient to see.