I have a music website where I embed music videos from YouTube. I plan to allow my website users to download these embeded videos using third party plugin for free. Does Google permit this or if I do so my domain will get banned from ever embedding their videos again or get banned from their Google search engine? I currently do not have AdSense on my site and I don't wish to put it.
Checking their Terms it is clear that YouTube do not allow downloads with this line, section 4 A:
You agree not to distribute in any medium any part of the Service or the Content without YouTube's prior written authorization, unless YouTube makes available the means for such distribution through functionality offered by the Service (such as the Embeddable Player).
However it is important to note that a company's T&C's are not law and there are situations where you are perfectly within your rights to do something not authorized in the Terms. For example, YouTube itself allows content creators to choose a Creative Commons license for their videos and so if a band released a video with this license, you could legally copy and redistribute it (usually with attribution required - read the fine print).
There are also fair use exceptions to copyright which permit unlicensed use of copyright-protected works in certain circumstances. Because you mention that you run a music website I'm guessing you would not be covered by those exceptions but the bottom line is that you are risking heavy fines if your activity falls outside the law. That should be your main concern, not whether YouTube wants to allow it or not.
One last point - definitely check out youtube-dl and its community of users if you have any more technical questions about how to download videos.