Hi John!
These are all very good questions!
I always say, comfort is most important on a Disney vacation. You're going to be traveling here to there so much, you want to be as comfortable as humanly possible :-)
Unless you're going to the more elegant Signature restaurants (some do have a dress code guideline), I would say where what pleases you.
You're going to find that Disney guests cover the entire spectrum of dressing: from business casual to T-shirts and jeans to the occasional Disney costume (yes, even some of the adults).
Running or jogging shoes I feel provide the best comfort, plus, it will save you from having to pack so many different shoes on your trip (although, your wife may have something to say about that :-)
Again, kids won't need to dress up unless you're going to the fancier restaurants (and even then, a number of kids and adults are in T-shirts and jeans).
As my mom would tell me, "If it's clean, not torn, and no holes. You're good!"
Have a great time on your trip.