Volume controller and screen size controller missing from Youtube app
There isn't a non-fullscreen layout for YouTube for 7in tablets, only for 10in tablets. There's no button there because there's nothing to switch to. The developers evidently think there's not enough room to see the video and display other information usefully at the same time.

You're right about the volume control. Because Android already has separate volume settings for media, notifications, and the phone ringer, and it guarantees the presence of hardware volume controls, it's considered bad design to put additional volume controls on the screen.