A "generation" isn't really a well-defined unit of measure. Generally, it's thought to be about 30 years for a given group, but it can vary; I've seen some define it as 40 years. If you mean lifetimes, that's going to vary by a lot of factors.

But to your second question, someone in the comments did the math, and I think it also answers your core question:

• The first gear takes about 3.5 seconds to turn.
• The second gear takes about 35 seconds, or 3.5x10^1.
• The fifth wheel will take around ten hours to turn once.
• In a month, the seventh wheel will have almost one rotation.
• The eighth will take a little over a year.
• If you watch this machine from the time you are born until the time you die, you will probably live to see the tenth gear make most of one rotation.
• The eleventh will take over a millennium to turn.
• The twelfth considerably longer than all of recorded history.
• The fourteenth wheel would take about as long as humans have existed.
• In the time since the dinosaurs went extinct, the sixteenth wheel would turn a little more than half way.
• Earth's existence has been long enough to get the eighteenth wheel half way around.
• In the entire history of the known universe, the twenty-first gear would move by just over one tooth.

ETA: all of these calculations are based on that initial 3.5sec rotation. If you increase RPM, obviously the machine will turn faster accordingly. There's some motors that can spin at 10k, 250k, and even 1M RPM.