Q1: It certainly has to give the same results, and it does so:

q0=0, q1=1, q2=0 yields

(0↔¬a∨Xq0) ∧ (1↔a∧Xq1) ∧ (0↔1∨q0∧Xq2)
= ¬(¬a∨Xq0) ∧ (a∧Xq1) ∧ ¬1
= 0

Q2: There is no need for a SNF, it is just suggested to compute the solution in the example solution. There are also many alternatives for sure.

Q3: For reasoning about SAT, there is no recipe that you can follow. If that would be known, you can win a million dollar price. Instead, you always have to check the formula and have to see what works best in the given case.