If the shliach tzibbbur starts saying kaddish on a day he is supposed to say tachanun, what does he do?
If the chazan already said the words “Yisgadal vyiskadash shmei rabba”, which would require the tzibbur to answer amen, then it is as if he started saying a bracha, and he has to continue finishing the kaddish. Regarding saying tachanun after he finishes kaddish, this would depend if this happened by shacharis or mincha. By mincha where the kaddish includes tiskabel, which is a requesting that Hashem accept our tefillos, then the tachanun has been disconected from the Shemona Esrei and it would not be said. However by shacharis, since it is only a half kaddish afterwards then tachanun can still be said after that kaddish.