Fullscreen youtube video on external display / avoid changing desktops

Unfortunately, you can't. I am hoping Apple will fix this as soon as possible as well.

There's a trick where you can put some app in the fabric space, as documented in OS X hints.

However, if you utilize the full-screen swipe gesture to move to a separate full screen and pause/stop in the middle with multiple Spaces in view and invoke another application (i.e. via Spotlight), the spawned application can reside in the 'fabric space' or anywhere else. It basically remains attached to that space.

Here's my workaround: right click on the 1/4-screen video and, if you're using Flash 15.0.0.xxx, you should see a Stretching is <type> item in the options list. Click on that and the video will fill the screen. Keep repeating until you get back to the 'Stretching is none' setting, unless you prefer something else.

I have the same issue when using firstrowsports.eu on my 2014 MBP, multiple different browsers, no plugins, no additional screen, hardware acceleration on and off, Mavericks and now Yosemite.