I found the "instruction manual" for you:
http://www.macobserver.com/tmo/artic...page-in-safari
What a horrible setup.

I warmly recommend to anyone to give those other browsers a test drive, instead of Safari.
As to the YouTube app, you're right that it's utterly unpredictable as to when it opens or doesn't open a video. That's connected to another major iOS flaw: inability to set default apps for specific actions/file types.
A major deficiency of the new YouTube app is that Google cancelled the back button in it!

That's right... if you watch videoclip 1, then videoclip 2, and would like to return to videoclip 1, it
cannot be done.

The
History menu inside YouTube is extremely buggy as well -- skips around 4 out of 5 clips I watch, so you won't be able to go return that way, either.
In short, YouTube is an awful app, and iOS is an awful OS... but it's what it is. Notwithstanding all the flaws, I'm convinced iPad Air is still the finest currently available tablet, so I believe you made the best possible purchase decision, especially having upgraded from a non-Retina device.