My sister recently received a talaq note from her husband; they have a 2 year old son. My question is: (1) Under Islamic Shariah who will hold the custody of the son, will it be the mother or the father?

Answered according to Hanafi Fiqh by

بسم الله الرحمن الرحيم

(Fatwa: 668/549/B=05/1435)

The boy shall remain in the custody of his mother up to the age of seven years.
قال فى رد المحتار: الحاضنة اما أو غيرها أحق به اى بالغلام حتى يستغنى عن النساء وقدر بسع وبه يفتى لأنه الغالب. (شامى: 5/567)
Note: The ruling shall be applicable when the woman does not marry to a man who is non-mahram for the child during his custody period.

Allah knows Best!

Darul Ifta,
Darul Uloom Deoband