Why all operators which we consider in CFT have fixed Dilatation value?
As I know in general QFT we haven't such requirement.
What if one will consider questions about operators, which are not Dilatation eingenvalues?
Actually the dilation operator is not necessarily diagonalizable, see https://en.wikipedia.org/wiki/Logarithmic_conformal_field_theory .
Also in the article On Scale and Conformal Invariance in Four Dimensions there's appendix about such phenomena.