What is the importance of calculating residence time of a Continuous Flow Reactor(CSTR) other than in reaction rate determination.

In a first look, I think residence time is also important in flowing characterization in order to detect by-pass or dead zones.
However, this use of residence time is more important in PFR/PBR reactors.

because you do not have an ideal condition, some materials will have shorter or longer period to reach the reactor outlet

Engineers seek to optimize yields. Chemical engineers also approach same issue another way in practice. They determine the theoretical heat output/input of the CSTR in terms of an easily measurable quantity like temperature and then track the difference between actual and theoretical temperatures. Residence time is one of several factors affecting yields from a CSTR.

you might get un completed reactions, intermediate products on one hand and at a second one - polymirazation/un wanted produfts; if the required products are left too long in a CSTR

Can we use a dye to calculate residence time of a real CSTR? With a step or pulse input. If the reactor is new and I use step input to calculate its residence time. Will the residence time of that reactor will be same after a year or two? if it doesn't then does this mean we need to clean the reactor from inside?

Dye tests will give you an indication of the residence time, but if it's mixing well working out the bulk residence may be difficult (dye will be very dilute). These tests may mean taking the unit off-line and/or damaging a batch.
CFD is a tool that will give you a good idea of the ideal/as designed flow patterns but won't tell you if the unit if fouling.

Residence time can be used also for calculating the right amount of cooling / heating power required by the reaction.

By theoretical definition, in a CSTR every point in the reactor has the same composition. So a step / pulse dye input wouldn't give you anything but an uniform colouring of your reacting fluids. What would you use the dye for? Maybe you can get to the "steady state" condition, then with a colorimeter you can derive the residence time from the colouring intensity but still depends on reagents / products / diluting medium (if any), etc. What is(are) the reaction(s) involved?

The tracer you can use will depend on what kind of laboratory analytical instrument or on_line analyzer you have in your facilities