Here's another vote for da UP route. Even if 6am departure didn't put you in Chicago during morning rush hour, I'd still recommend avoiding it. Stop and go traffic, and the stress that goes along with the extra vigilance needed by the driver, is best avoided... especially on the first day of what should be a fun trip. The UP route is an extra 30 or 40 miles, but miles well spent IMO.
I have read many past suggestions from experienced Yellowstone campers, saying that it's better to divide YNP into 2 or 3 sections and reserve a couple nights in each, due to the crazy drive times.
You ought to allot one day to drive the Beartooth, US212. Everyone says it's fabulous, and I would like to see it myself the next time I get to the area. It's pretty easy to get to from Cody.
I can't say whether the bikes or kayaks would come in handy in YNP, but they would in the Tetons. After doing a bunch of driving and hiking, it can be a nice change of pace to paddle or pedal for a while. But the water is very cold in these mountain-stream-fed water bodies, so be prepared (maybe some neoprene tops?). There is a nice multi-purpose trail from Jenny Lake southward for your biking. If packing these items would be a hassle, just rent 'em when you get around Jackson Hole.
Be sure to allot a few hours for a leisurely drive along the winding roads and through the tunnels at Custer SP. It's a great place. Expect plenty of wildlife, too.
If you head south to I-80 for the trek home, stop at Vedauwoo (just off the highway, east of Laramie) and let the kids clamber around on the rocks.
