Can Flex thrust be lower than climb thrust?
Yes.  On the CF-34 family, in theory, a maxed-out flex setting can be in the mid 80s N1 whereas climb thrust is normally high 80s/low 90s, and you could see thrust rise when setting climb if an extreme flex setup (max Assumed Temperature) (or in the case of the non-fadec engines like the 3B1, the climb bug appears at a higher value than the flex take-off thrust bug when you select "climb" thrust in the FMS)

The reason is to do with what you might call the mission priority of the two settings, and where the actual benefit in flex lies.  

With flex, the priority is distance and gradient for obstacle clearance.  The flex setting is the lowest setting that meets the takeoff and initial departure performance requirement for accelerate/stop and initial climb gradient for that runway and all up weight.  The benefit of using the lowest setting is only partly from lower temperatures; an even bigger factor is erosion from silica particles (outdoor dust) which is most severe below 1000 ft.  The dustier the environment, the larger the flex benefit. (In the Middle East, CRJ operators were wearing out engines after only 5-7000 hours because they were doing max thrust takeoffs all the time and had to be cajoled into using flex.)

Climb thrust has a time priority, from an operational balance-of-factors perspective.  You want to get to altitude in an optimized time-related profile (as soon as possible taking various things into account), and an optimized time-related profile can require more thrust than the minimum required to meet take-off performance, in cases where a very large flex allowance is available allowing minimal take-off thrust.  Setting climb thrust that is a little higher than the flex take-off setting has little negative impact on engine life because you are now high enough that the erosion impact is negligible so there's little engine life benefit to a slower climb to altitude and you might as well optimize climb thrust to get there faster.