Do we say tachanun after selichos recited at night ( before chatzos)? If yes, please explain the reasoning. Thank you.
No, Tachanun is not recited at night, certainly not before Chatzos. After Chatzos the matter involves a dispute among authorities.
See Otzar HaGeonim Megillah 22b; Tzeror HaChaim p. 18; Mahari Abuhav as cited in the Beis Yosef 131; Shulchan Aruch 131:3; Mishnah Berurah 131:17-18 citing from several Acharonim. According to the Shulchan Aruch HaRav one does not recite Tachanun until after dawn.