What are the consequences of solving XOR 3-SAT in Logspace?
Take a look at https://www.sciencedirect.com/science/article/pii/S0022000008001141 "The complexity of satisfiability problems: Refining Schaefer's theorem" by Allender et al. which answer your questions:

(1) Open

(2) $\oplus L = L$

(3) Yes