This part is fine.There's no need for special rules in the knowledge skills to cover these situations, because they're covered in the rules for all skills. It explicitly says that if you're just walking down the street, you don't need to roll balance to not fall over.
This part, however is not. What you should and should not be able to do should be based on what the modifiers add up to allowing you to do, not based solely on what the DM thinks is acceptable. If you want the Rogue with 18 dex and balance trained (+7 modifier) to be able to succeed reliably, that check DC needs to be 17 or lower. That means your Wizard with 8 dex and no training (-1) can succeed on that check 15% of the time. Telling him he can't even roll to attempt it is robbing the player of that chance without any mechanical reason for it.Similarly, if a clumsy (8 dex) wizard tries to walk across a thin, rotted rope strung between two ships being tossed in stormy seas, you don't roll for that, the DM just tells you how badly you fail. The same rule applies when your dull (8 int) rogue tries to recall the jewelry being worn by the Djinni involved in the ancient conspiracy between the Elemental Plane of Fire and the Elemental Plane of Slightly-Less-Fire that occurred three thousand years ago.