Let me just address 1. for now. I just followed the prescription in my Vol I book, pg 33. That is, the characteristics [$]\xi[$] are the solutions to the ODE:

[$] \frac{d \xi}{dt} = b(\xi) \equiv (1 - \xi)^2[$] for this problem, with [$]\xi(t=0) = y[$].

Since the solutions [$]\xi(t)[$] also depend upon y, write them finally as [$]\xi(y,t)[$].