Whereabouts on the Kohen Gadol's head was the Tzitz worn?
Refer to 28:37:2:1 & 2.
Targum Yonasan (on Pasuk 37): It was worn on the Mitznefes above the Tefilin shel Rosh. 1
See Na'ar Yonasan.
Which sin did the Tzitz atone for?
Rashbam: It atoned for any sin on which one brings an Olah, a Chatas or an Asham to atone - together with the respective Korban.
Targum Yonasan: It atoned for the abusal (Me'ilah) of Hekdesh. In Pasuk 37 however, Targum Yonasan exlained that it atones for Chutzpah. 3
Erchin, 16a: It atones for Chutzpah. 4
Not for Kohanim who brought Korbanos (or who ate ate Kodshim - Rashi Pesachim, 16b) when they were Tamei.
And not for the sin of Pigul or Nosar (See Rashi). Because Tum'ah has the leniency that it can be brought be'Tzibur (Torah Temimah).
Which explains why it is worn on the Kohen Gadol's forehead ('Metzach' - since Chutzpah is also known as 'Azus Metzach') See also Peirush Yonasan. By the same token, the Gemara in Erchin (16a) points out that each of the Kohen Gadol's Begadim atoned for a specific sin ... The tunic for murder, The pants for adultery, The hat (turban) for arrogance, The belt for immoral thoughts, The Choshen Mishpat for perversion of justice, The Eifod for Avodah-Zarah, The Me'il for Lashon-ha'Ra and The Tzitz for Chutzpah. See Gemara for the sources. (EC)
What if the Kohen Gadol was not wearing the Tzitz?
Yoma, 7b: The Tzitz only atoned whilst the Kohen Gadol was wearing it, as implied by the Pasuk. 1
This does not create a problem with the occasions on Yom Kipur when he only wore the four Begadim
What are the implications of "le'Ratzon lahem"?
Yevamos, 60b: It implies that the Tzitz will not respond to anything punitive concerning Yisrael ("lahem"), but it will respond to the same request if it concerns Nochrim. 1
Zevachim, 45b: It implies that the Tzitz only atones for a Yisrael (who brings a Korban be'Tum'ah), but not for Nochrim, who are not subject to the goodwill of Hashem.
Such as a question whether to kill a Yisrael or not. See Torah Temimah, note 27.
Bearing in mind that the Kohen Gadol only wore the Tzitz whilst performing the Avodah, why does the Torah insert the word "Tamid"?
Rashi #1 (citing Yoma, 7b): To teach us that the Tzitz atones even when the Kohen Gadol is not wearing it.
Rashi #2 (Ibid.): To instruct the Kohen Gadol to constantly touch it, in order not to take his mind off it. 1