I have a "Harness and Vehicle" parts list on my webpage that I specifically posted for newbies.
http://rhinestone-ridge.wikispaces.com/Cart+and+Harness+Parts That might help you with the terminology. There are some parts with acceptable multiple names for the same part, and I have some of those listed, but what I have listed is acceptable for the carriage/light horse driving community. Draft drivers might use different terms, as well as other disciplines.
Bicycle wheels are not acceptable in any circumstance. Here is what can happen to a wheel on even just a trail.
This was taken after a Distance Drive, which doesn't necessarily consist of hard turns and stops. I'm not even sure this is a bicycle wheel, it might have been an actual cart wheel, but obviously it "didn't make it" through the whole drive. Just like there is "no hoof, no horse", there is also "no wheel, no cart". I can't stress this enough, USE GOOD WHEELS! I have more photos of other wire wheels that "didn't make it".
As far as the size goes, basically, the shafts need to be level or just slightly uphill, but never downhill. So you won't know this until your horse is done growing. You need to measure your horse from the point of the shoulder to the point of the buttock. Then add about 8-12" depending on the height of the basket on the vehicle (you don't want your horse to hit his heels on the cart). I like my horse closer vs. farther from me, but they definitely can't hit their heels!
I just learned from an FEI driver at a clinic that the walk is actually the "longest strided" gait, so if they don't hit their heels at a walk, they probably won't at a trot no matter how big the stride. Generally, the horse's hind legs go under them more at a trot, hence the walk being "longer".
Welcome to the driving community!
Myrna