Dragging a tab tears it out of its current window to a new window, so that isn't going to work.

If you drag the favicon (site identity icon) from the urlbar, you get the URL and page title, and the receiving application decides what to do with it. I don't think any page content is provided to the receiving application.

Perhaps you should consider creating an add-on or userscript to extract the information you want and load it on the clipboard, or export it to some kind of storage? I'm not sure you can get information from the Flash player plugin, if the page is using it. (For that matter, I don't know whether you can get the current video time from the native HTML5 video player, either.)

Okay, well in the past I've bookmarked a Youtube page mid-video, and upon re-opening the bookmark I'm brought to the exact same frame in the video.

Do you understand why that would be?

It's a extra command added to the end of the Youtube url. You have to add &t=*m*s to the end of the url. Where the asterisks are, you replace them with numbers. For example: http://www.youtube.com/watch?v=Zb7gTeLmpHs&t=3m24s

The m is for minutes and s for seconds.

You can also find this information by loading a Youtube video, dragging the time slider to the time you want, pause the video, then right-click on the video and choose:
"Copy video URL at current time".