There are quite a few questions there!
One subtle answer that I know, because it was the subject of considerable analysis a couple of years ago, is that the best rate of climb at any point on the climb profile does not always give the quickest time to a given altitude (or the lowest fuel burn) - at least for the Citation family, but probably other types too. I suspect this corresponds to your question number 4.
Suppose you launch from sea level, aiming for FL400, and try to get there as quickly as possible. Intuitively you might expect that if you carefully maintain the best-rate-of-climb airspeed all the way up, you would get there in the shortest possible time. Not true! You will actually get there quicker by climbing at cruise climb speeds! The analysis needs large volumes of performance data to explain fully, but the simple pilots explanation is that at the lower altitudes you might as well fly faster, cover more ground, but climb a little slower, and BURN MORE FUEL. At the lower altitudes, you've got lots of power to spare. At the higher altitudes, your margin is very modest, and arriving there lighter gives a marked improvement in climb performance. If you rush to high altitude, you'll reach a point where the rate of climb is very poor (because you are relatively heavy) but also the fuel burn is low (so you are stuck there). So, the "max rate" pilot might have spent about 5 minutes of the climb in the bottom half of the atmosphere, and will spend the next 30 minutes climbing the rest of the way. The "cruise climb" pilot may have spent 10 minutes in the bottom half of the sky, and will spend 15 minutes in the top half, and will actually get there before you. Obviously these numbers are illustrative only, your mileage may vary, etc.
If you want to try the calculation for your own favourite bird, you'll need the flight manual, a monster spreadsheet program, and a deal of patience to enter all the performance figures.
As to whether your chosen FMS is clever enough to do that in real time - hmmm...
