2. Parents have prophecy to name a child: Therefore, the AriZal (Rabbi Isaac Luria) writes. "When a person is born and his father and mother give him his name...the Holy One puts into their mouth the particular name required for that soul" (Sefer HaGilgulim, introduction 23; Emek HaMelech, Shaar 1 end of Ch. 4; Or HaChayim on Devarim 29:17) (Igros Kodesh Vol 7 page 144 and 308, Vol 9 page 136)
3. The source of giving a deposit for schar limud (tuition for Torah education) when a child is born is from the Frierdike Rebbe, Hayom Yom 28 Adar II:
"... It is our custom to make an advance payment on tuition fees for the boy's studies. (Here the Rebbe gave a sum of money and said): This is for the Yeshiva...."
4. As per Hisvaaduyos 5744, vol. 2, p 827 a letter for the child in the Sefer Torah should be done as soon as possible after birth, even before naming the baby. The baby's name can be added afterwards.