MQM90, I'm a little rusty on these things, but:
a) Cycles used must be based on a realistic evaluation of the operating conditions. If a vessel is started and stopped once a day, 1000 cycles won't get you much life.
b) Thickness of a F&F joint will be dependent on the applied stresses versus allowables. However, as the joint gets thicker, the spring rate also increases such that the joint may not prevent overloading of the tubes or tube-tubesheet joint. A good general rule of thumb is that if you don't really need an expansion joint, a F&F will work just fine.
c) See above for thickness versus applied stress versus allowables. Often client specs will not allow multi-ply joints.
d) The best joint is NONE. However, this is not always possible and so the selection will really depend on the differential expansion and to some extent design pressure. Very small amounts can often be accomodated by a F&F. Anything over about 1/8" will most likely require a bellows joint.
Some other pros and cons are that you generally have to design a F&F yourself, and then find a vendor to make the design, for which they will propose something a little different dimensionally which you then get to re-analyze. Repeat as required.
For a bellows joint, you can furnish the manufacturers with the design conditions and other desired features of the joint and they will design it. You then "only" have to review the documents, insure delivery is as promised, etc, etc, as when using any vendor.
A F&F joint is very rugged, however a bellows is very much less so and needs to be treated delicately. Bubba can take one out with his big rodeo belt buckle, no offense to Bubba intended. It can blow up at hydro with no warning, possibly requiring the services of the manufacturer to repair it on an emergency basis.
Note that my remarks are not related to a particular code of construction as you did not mention one, although my experiences are with ASME Section VIII, Div. 1. You would want to carefully read your actual code used, if any.
This is all kind of general, perhaps some of the other participants can give better information for you.
Regards,
Mike