The use of a FLEX thrust setting for takeoff on IAE engines will generally result in a thrust reduction when selecting climb thrust. At max FLEX temp it may be very close to climb thrust, I couldn't say if it can be lower.
With respect to climb thrust, as long as the pilot doesn't modify perameters in the MCDU, climb thrust is determined automatically from entered temp, weight, cost index etc and it is then selected in flight by placing the thrust levers into the climb detent. Pilots don't generally change thrust settings at all after that.. It's all computer driven.
If required it is possible to increase thrust beyond that by selecting MCT or TOGA