Why does the Torah insert the word "Zeh", which always implies something visible?
Rashi and Targum Yonasan: Because Hashem showed him a coin of fire 1 to demonstrate exactly how the half-Shekel should look.
Hadar Zekenim: They descended [at the Egel] with Zeh -"Zeh Elohecha" (Nechemyah 9:18), or "Zeh Moshe
Da'as Zekenim, Hadar Zekenim (12, citing Tanchuma): He showed him with His right thumb. All the fingers of His right hand are connected with Ge'ulah. Also see Ba'al ha'Turim.
Rashi writes that Hashem showed Moshe a 'coin of fire.' Why was this needed?
Hadar Zekenim (12), Moshav Zekenim (25:8): Three 1 things shocked Moshe - one was how such a small donation could attain atonement for the soul (until Hashem showed him a coin of fire).,
Michtav me'Eliyahu (5, p. 417): If one must redeem himself with quantity, there is no limit how much is needed to redeem sin! Hashem took the Shekel of fire from under the Kisei ha'Kavod (Tanchuma 7), to hint that pure zeal for Hashem's honor, amidst Hashem's Chesed, a little suffices - "open for Me like a needle's eye
Hadar Zekenim, Moshav Zekenim: The others were (a) how Hashem, whom Shamayim does not contain, could dwell in the Mishkan, and (b) why He commanded only two lambs a day for the Tamid.
What is the meaning of "ha'Over Al ha'Pekudim"?
Rashi: When one counts, it is customary to pass whoever (whatever) is being counted before the one who counts. 1
What is the significance of the half-Shekel?
Da'as Zekenim, from Yerushalmi Shekalim, 2:3): It is because [the root of Chet ha'Egel] was at midday (when Moshe did not return - Shabbos 89a).
Torah Temimah, citing Shekalim, 2:3: Because they transgressed the Ten Commandments (and half a Shekel equals ten Geirah). 1
Da'as Zekenim, from Shekalim, 2:3: Because [the root of Chet ha'Egel] was after six hours, and half a Shekel equals six Garmisin (a coin of that time).
The Gematriya of Shekel is Nefesh. Hashem in His mercy atones for the sins of half the Nefesh, so a half-Shekel atoned for the other half.
See also K'li Yakar.
What is a Geirah?
See Shmuel 1, 2:36. Initially, a Zuz comprised five Ma'ah, but Chazal added one sixth (a fifth of the initial value), to make it six (Rashi). See also Ramban (on Pasuk 12), who cites a Pasuk in Nechemyah that they gave a third of a Shekel (ten Geirah) - because Ezra raised the value of the Shekel.
And one Shekel equals four Zuzim (Rashi).
What is the significance of "Shekel ha'Kodesh?
Rashi: It is the Shekel that Hashem fixed to use for Divrei Kedushah, such as the Shekalim mentioned in Bechukosai (chapter 27) in connection with Erchin and Sadeh Achuzah.
Ramban #1: Whatever emanates from the Torah bears the title Kodesh. Consequently, the language of the Torah, with which Hashem communicated with the prophets and with Yisrael 1 is called Lashon ha'Kodesh, And by the same token, all sums of money mentioned in the Torah (which are all in terms of Shekalim) are called Kodesh.
Ramban #2 (citing Rambam in Moreh Nevuchim): The language is called Lashon ha'Kodesh because there are no intrinsic words for the Ivarei Tashmish of a man or a woman, nor for a drop of Keri, urine or excrement - they are all presented with descriptive titles which convey their meaning. 2
Ramban: Moreover it is the language by which Hashem Himself is named (Keil, Elokim ... and Havayah), and it is the language which He used to create the world, and with which He named Heaven and earth and all their hosts, as well as all the angels (Micha'el, Gavri'el).
See Ramban's objection to this explanation.
How much is the Shekel mentioned in the Torah?
Why are there two notes to read the word "v'Nasnu", Kadma v'Azla?