Recently I saw in Halacha that it is inappropriate to have the sermon right before Kaddish of Musaf as it is considered a Hefsek. Yet many Shuls do just that and have the sermon right before Kaddish of Musaf. Is there any justification to do that? What is the source?
Although the Taz (S.A. O.C 55:3) only requires a minyan at the time of kadish, the Magen Avraham (69:4, 234:1) and others say that there needs to be a minyan at the time of pesukim/learning for the kadish to go on.
I assume that the shuls don't want to leave the Torahs out and don't want to change the order of davening (and say ashrei after the Torahs are put away), and so rely on the pesukim in the drasha. Perhaps even the Taz (234:1) would not consider it a hefsek because the drasha is part of the pesukim.
When one puts on Rabbeinu Tams before musaf on Rosh Chodesh, the custom is for the Chazan to say a pasuk quietly before Musaf because the donning of Tfillin is a Hefsek (though those not wearing Rabbeinu Tam Tfilin don't because just taking of tfillin is not). Perhaps that is what one should do (see LOR!).
Though many say it is also disrespectful to speak before Torah reading (It's like saying "My speech is more important than the Torah").
Speeking after Davening isn't possible because people would run away before.