Is the giving of a penance by the priest at confession required, and is the performance of it necessary for the sacrament to be valid?

The priest is required to impose a penance, or satisfaction on the penitent. It should be helpful, prudent, just and suitable based on the kind and number of sins, and on the character and the condition of the penitent. The priest is only excused from imposing a penance when there is some physical or moral inability on the part of the penitent to perform it, e.g. if the penitent is near death, or too weak.
 
The penitent has a serious obligation to accept and fulfill the reasonable penance imposed by the confessor. If a penitent considers a penance unreasonable they are free to ask for another penance from the same or a different confessor.
 
The giving and fulfilling of the penance does not per se affect the validity of the absolution that is given, unless perchance the penitent approaches confession with such a determined will to refuse any penance that it affects the necessary contrition he must bring.
 
More often, the failure to give or fulfill a penance is due to forgetfulness, and in such cases the validity of the absolution is not affected. If however, the giving of or the fulfilling of a penance is intentionally neglected, while the validity of the absolution may not be affected, one does incur sin. The gravity of that sin is weightier if the material of the confession was grave, or serious.