WIY: For whatever it’s worth, I also say Tachanun before Chazaras Hashat”z. I don’t do Nefilas Apayim, however, because that would be an Issur D’Oraisa of Lo Sisgod’du, even if I think the shul’s Minhag is a Minhag Ta’us. There’s no Issur of being Mafsik between SH”E and Chazaras Hashat”z. However, one should not interrupt between SH”E and Tachanun (see the Aruch Hashulchan who speaks very strongly on this).
