When exactly is "mi'Ben Chodesh"?
Rashi (in Sh'mos 22:28): It means after thirty days.
What is the significance of the thirty days waiting period?
Shabbos, 135b: It teaches us that a baby only leaves the realm of Safek Nefel after thirty days. 1
Bechoros: We learn via a Gezeirah Shavah "Chodesh" "Chodesh from "P'kod Kol B'chor Zachar ... mi'Ben Chodesh Vama'alah" (Bamidbar, 3:40) that the obligation to redeem his son begins after thirty days, and that if he paid the Kohen during the thirty-day period and the baby died, the Kohen is obligated to return the money.
See Torah Temimah, note 57, who elaborates.
Can a father redeem his firstborn son with something other than money?
Bechoros, 51a: We learn via a K'lal u'Perat u'Kelal: "u'Feduyav" - K'lal; "be'Erk'cha Kesef Chameishes Shekalim" - P'rat; "Tifdeh" - K'lal 1 that one may redeem with anything that can be moved and that has an intrinsic value (like money) - to preclude Karka, which cannot be moved and Avadim - which the Torah compares to Karka - and documents - which have no intrinsic value.
See Torah Temimah, note 59.
How much is twenty Geirah?
Refer to Vayikra, 27:25:2:1 and note.
Are Beis-Din permitted to add to the value of the Shekel or subtract from it?