Answer to Question 1
e.
This statement is not true. Goudsmit and Uhlenbeck proposed that the electron have an intrinsic spin quantum number of s = 1/2. Schrdinger's equation fully solved did not include this quantum number.
Answer to Question 2
The Coulomb potential field that is approximated as the only one affecting the electron in the cases with one electron around the nucleus gets far more complicated with the addition of even one more electron. The Schrdinger equation becomes very difficult to process.