During Kaddish one is not allowed to talk (duh) but . . . one must stand silently and listen intently to the words and answer the appropriate responses.
Sfardim do not stand for Kaddish (unless they are saying it).
A shul is NOT allowed to skip tachnun when there is a bris earlier that day in the same shul . . .
It depends on what the Rav of the shul says. Our Rav says all minyanim skip Tachanun on the day a brit is held in the shule. I have never asked him his sources for this, but since he knows halacha forwards and backwards, I trust he has sources to rely upon.