Well done williamhj.
I can add a few things that were not addressed.
"1. it would seem to me that a 7wt will certainly be able to cast a fly that is too heavy for a 3wt. At least it seems it would be able to cast it better: more accurate and farther."
A 7 weight will be able to cast farther but not more accurately. Accuracy is not a function of the line rating but is affected by fly rod length as described below.
'5. Finally, does the length of a rod only limit the casting difference? For example, if you had an 7-foot 5wt, you may not be able to cast as far as the same rod that is 9-feet long, right? Or are there different reasons other than simple distance as to why you would use a longer or shorter rod? (Well, you use a shorter rod whenever you fish in constricted areas--are there other reasons?)'
A longer rod is a longer lever, and longer levers magnify the motions of the caster's rod stroke. So a longer fly rod is able to move the rod tip over a greater distance for a given rod stroke. A longer rod stroke translates to a longer cast, when all other factors remain equal.
See #4 in the illustration below:
This lever effect also means that the caster cannot control the rod tip motion as easily as in a shorter fly rod. It is like trying to write your name with a paint brush that is 4" long vs 8" long while holding the end of the brush. So longer rods move the rod tip over greater distances but with less precision, again if all other factors are equal.
That means that the shorter fly rod is better for accuracy within the nominal range of that rod but will not be able to cast as far as the longer fly rod.