Can someone say Tachnun for sharit or mincha when praying alone at home? Can he or should he place his head on his arm or not without a safer torah present like in a shul?
Tachanun is recited even when an individual davens alone, with the exception of theselichos part of tachanun (which is recited by Sefardim daily, and by some Ashkenazim on Mondays and Thursdays).
However, one does not “fall on one’s face” when davening in the absence of a sefer Torah, with the exception of Jerusalem, where many poskim write that one says tachanun with nefillas apayim even in the absence of a sefer Torah.
See Shulchan Aruch and Rema 131, and Mishnah Berurah 131:11-15.
Concerning Jerusalem, see Ishei Yisrael ch.25, footnote 39, and Ve-Aleihu lo Yibol vol. 1 pgs -96, 298, 348, in the name of Rav Shlomo Zalman Auerbach, who stated that this only applies in the Old City. However, others, including Iggros Moshe (Orach Chaim vol. 3.129,2), Beis Baruch (32, 170), Minhagei Eretz Yisrael (Tukachinsky), and Teshuvos Ve-Hanhagos (vol. 2,79; vol 4, 41) do not make this distinction, and write that the halachah applies to the whole of (the extended) Jerusalem [these sources were provided by R. Yehudah Spitz].