The YouTube app is not included in iOS 6. You will either need to try one of the YouTube apps from the store (there is an iPhone optimised app from google, and other people have said that they are working on an iPad version), or try accessing the mobile version via Safari : http://m.youtube.com (you can create a home screen shortcut for the site)
For sounds, have you got notifications muted ? Only notifications (including games) get muted, so the Music and Videos apps, and headphones, still get sound.
Depending on what you've got Settings > General > Use Side Switch To set to (mute or rotation lock), then you can mute notifications by the switch on the right hand side of the iPad above the volume switch, or via the taskbar : double-click the home button; slide from the left; and it's the icon far left; press home again to exit the taskbar. The function that isn't on the side switch is set via the taskbar instead : http://support.apple.com/kb/HT4085