iOS 6 removes the YouTube app from Apple, but you can still watch YouTube videos in the Safari browser via the YouTube plug-in (without need of Flash). If that is too slow, you can get the YouTube app from Google on the App Store. If that app's interface is too buggy, my personal favorite is an app called Jasmine. It too plays YouTube videos, but has a great interface similar to the old iOS YouTube app.
You don't upgrade YouTube. It will do system updates and change format etc., by itself.
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it gives you air play and makes it harder to jailbreak
You can't.
Yes. If you update your iPhone's software using iTunes it will lock your iPhone again. You will then need to unlock it again.
plug your iphone into your computer and go into itunes click on your phone under devices go to summary and press check for update or update iphone.
Simple answer: Yes.Only update when the Jailbreak you use has a new version / then only update the Jailbreak, not the iPod/iPhone.
No.
If you are certain your wireless network is working properly and you can use Safari with no problems this may be, as you say, a bug within the YouTube app. If this is the case there isn't anything you can do but wait for a fix to be released by Apple in the form of an iPhone OS update. This may be fixed with the iPhone OS 4.0 software update being released in the summer.
The exact release date is 17th June 2009. To find out more go to apple.com/iPhone and you'll see info about iPhone OS 3.0 and the new iPhone 3G S. Which is the new iPhone with voice recognition and video recording.
Upgrading an iPhone will update the firmware (the iOS) and will therefore remove the jailbreak. If you wish to stay jailbroken, it is highly recommended to not update until news of a new jailbreak arrives for the upgrade.
The update 6.1.6 helps to maintain the working efficiency for the iphone 3gs.