I'm not a lawyer but you really can't add any videos to your website as "fair use" You ALWAYS have to get written permission.
The copyright owner has to give you permission to republish the video or deny you the permission. If the owner wishes, they can also allow you to republish the video only if you pay them certain royalties.
There are also instances where you should NOT embed a YouTube video. If the video you see is of some commercial TV show or movie, it is not wise to embed it. While it is of course possible that the commercial TV or movie company put up the video themselves and want others to embed it, there is also a chance that someone other than the copyright owners has illegally uploaded the video. Embedding such videos puts your site (and you) at risk. You may have the most innocent of intentions, but if the copyright owner takes you to court, you'll end up spending money, wasting time, worrying unnecessarily, just to prove your innocence. It's simply not worth it.