Does the emission spectrum for a dye depend on the frequency of light used to excite the dye?
Does the emission spectrum depend on exciting the dye at its peak absorbance wavelength $\lambda_{max}$? 


No, it is independent of that. The excitation $S_0 \rightarrow S_1$ is vertical, bond length are not changed in the excitation process. Compare the mass of the nuclei with that of the electrons and you know why. You'll just end up in a different vibronic level of the electronically excited state. 

However, before the emission takes place, this initial Franck_condon states relaxes to the vibrational ground state of $S_1$ with an equilibrated geometry and emission occurs from that (Kasha's rule).