Changing thrust does not change minimum control speeds - they are based on full-thrust.
I guess that depends on the implementation. On the aircraft i flew that had derates available, or user selectable thrust settings (737NG), the minimum control speeds are based on the chosen thrust setting, but higher thrust is still available. So yes, using thrust that would lead to loss of control at the minimum control speeds for the lower user selectable thrust setting is very possible.
Using that kind of derate has some positive sides, it allows to get higher loads out of contaminated runways and allows bigger thrust reductions than assumed temperature alone.
However, increasing thrust after an engine failure would be a very bad idea indeed if the speeds used are close to or at the minimum control speed of the derated thrust setting.
and thus CLB cannot be more than that. Or can it?
It can. Well, especially so if one uses a higher climb thrust than the corresponding reduced climb thrust. But even if one uses the corresponding reduced climb thrust it can be higher than the reduced take off thrust at light weights, but usually only if one combines a derate 2 and assumed temperature.