Can someone share some good scholarly references where the left and right inverse of functions are defined, and their properties analysed?
Even better if they are references that also discuss various different definitions in circulation, their advantages and drawbacks, and their different usefulness in different branches of mathematics.
The google search "mathematics teacher" "function" "codomain" produces several items of possible interest to you, such as:
The mistakes that are made by students with regard to functions: evidence from Erzincan by Kani Başibüyük, et al. (2016 math education paper; 10 pages)
Undergraduate mathematics students’ understanding of the concept of function by Caroline Bardini, et al. (2014 math education paper; 23 pages)
The Function Concept and University Mathematics Teaching by Olov Viirman (2014 Ph.D. dissertation; 99 pages)
Teaching Students to Communicate with the Precise Language of Mathematics: A Focus on the Concept of Function in Calculus Courses by Derrick S. Harkness (2020 Ph.D. dissertation; xii + 137 pages)
Also of possible interest are some of the references I give in this answer, especially The rise of functions by Salomon Bochner as it focuses more on modern ideas than the other papers.