Is it possible to have live video streaming on my website but using a YouTube line (embedded)?
So the YouTube logo will still be on the video but people will watch it live on my website.
Do you mean by using Google Hangout? Then the answer is yes. It is similar to embedding a YouTube video. Once you have started airing your video, Hangout generates a YouTube link and the embed code which can be used to display it live on any website.
See this video: http://www.youtube.com/watch?v=PdXogKQTVbE
It is for WordPress but you can do similar things in your website.
Yes. It is completely possible.
Just embed your video onto your site. Add in the HTML code, and if you need help doing that, just post another question here.
YouTube live is only available to people who it has been given to, and if you're not talking about another's live video or your own, please rephrase your question.