Cut-off wavelength for absorption study is used for the solvent not the dye. The idea is, if your solvent absorbs in the same region as your dye a quantitative measurement will not be possible. So proper solvent is chosen so that it does not interfere with the dye. Let's say somebody is interested in measuring the absorbance of benzene in DMF. Since DMF has as solvent cut off of 270 nm we won't be able to get proper spectrum of benzene. So cyclohexane (cutoff of 200 nm) should be chosen as a solvent for benzene. In your case DMF or CHCl3 does not interfere at all.  In-fact any solvent (even acetone) should be fine for your dye. If you are using your dye as a solvent/medium/matrix then the cutoff should be around  500 nm.

https://www.researchgate.net/post/What_is_the_UV_cut-off

I found this in someone's thesis. He used Cut-off wavelength to find the optical band gap energy. 

Band gap is applicable where there is valence band and conduction band. Organic molecule do not have such. For organic molecule generally the peak is considered the energy gap. More specifically the overlapping wavelength  (if there exists any overlap at all)  between absorption and emission spectra is called pure electronic energy gap.

Tauc plot is used to calculate the band gap of materials. I am not sure whether cut-off wavelength is the commonly used term here. Please go through the links

http://onlinelibrary.wiley.com/doi/10.1002/pssb.201552007/full

http://www.appropedia.org/Tauc_Plot_-_Determining_Optical_Band_Gap

Here is an excellent discussion regarding Tauc plot

https://www.researchgate.net/post/How_to_draw_Tauc_plot_from_absorbance_spectrum_for_ZnO_nanoparticles/1

I am sorry for late reply. Thank you for the explanation :-) It was really helpful.