tefilla - Does the congregation recite certain sections of the prayers if a mourner is in the synagogue?

This morning, a mourner (in Shiva period) appeared in my shul.

Art Scroll Ashkenaz siddur, among others, most likely, indicate that in a house of mourning, Tachanun is omitted and in Uva Letziyon, the phrase Va'ani Zot is also omitted.

I'm not sure if these rules are specific to when prayers are done in a mourner's home (i.e., does it depend on the location?) or would they skip these items in a synagogue where the mourner is present (i.e. it is based on the person)?

Additionally, if the mourner is the Shaliach Tzibbur (cantor), even if the congregation says it, should he, personally skip these?