To what extent is it permissible for the muezzin or the Imam to say, after the Iqamah: "Stand close together, and you will be shown mercy" (تزاحموا تراحموا)? And is this a principle, a Hadith, or an Athar (statement from a companion or successor)?

The phrase "tazahamou tarahamou" (crowd together, have mercy on one another) is not a marfu‘ hadith (definition) (a narration directly attributed to the Prophet), nor is its origin known from the Salaf (pious predecessors). It should not be adhered to. Crowding for the first row is permissible due to its virtue, but it must not lead to harming or inconveniencing other worshippers. Rather, if a row is full, the worshipper should stand in the next row. Warding off harm from a Muslim takes precedence over performing a sunnah (definition) (recommended act), based on the Almighty's saying: "And those who harm believing men and believing women for [something] other than what they have earned have certainly borne slander and manifest sin." Therefore, abandoning a sunnah to prevent harm is better than performing it while causing harm.