Larry, your post alludes to the correct answer to Q1. There is also a non-algebraic way to think of it.
Your calculation of the times at which they overlap is nicely set up (which is very difficult to do) and is correct up until the very last step:
So for example, to find the time X when the minute hand and the hour hand first coincide, starting from 3:00am, the minute hand will travel 15+X minutes while the hour hand will travel X minutes. The equation can be written as
(15+X)/(60m/h)=(X)/(5m/h), or (15+X)(5m/h)=X(60m/h).
So we get 75+5X=60X, or 75=55X, or X= 75/55 = 15/11 = 1 4/11 minutes after 3:00am.
On my clocks, the hands do not align at approximately 3:01:22. But at 1 4/11 minutes after 3:15 they do! (Which is of course what your equation shows.)
Thoughts on Q3?