There's no way that your "solution for the PDE with another method" is correct.

Your equation can be rewritten as $$ u_y = \frac12 (x - u_x^2) $$ Evaluating on the set $\{x = 0\}$, you have $$ u_y(0,y) = - \frac12 u_x^2(0,y) \leq 0 $$

On the other hand, your initial data has $u_y(0,y) = -2y$, which is strictly positive when $y < 0$.

Your mistake is in the part you didn't show in the question.

If you are interested in only solutions on subdomains, near the set $\{x = 0, y > 0\}$, then your mistake is that in the method of characteristics, the fact that $q = a$ only holds along the characteristic. (Your supposition that $q = a$ everywhere already contradicts the initial data.)

Everything you did after that is not part of the method of characteristics.

Given $$ F = p^2 + 2q - x $$ The characteristic equations are $$ \dot{p} = 1, \dot{q} = 0, \dot{u} = 2p^2 + 2 q, \dot{x} = 2p, \dot{y} = 2 $$ You obtain that $$ p(s) = s + p(0), q(s) = q(0), u(s) = u(0) + \frac23 s^3 + 2 s^2 p(0) + 2p(0)^2 s + 2 q(0)s + u(0)$$ and $$ x(s) = s^2 + 2p(0) s, y(s) = 2 s + y(0) $$ The initial data are given in terms of $y(0)$: $$ q(0) = - 2y(0), p(0) = \pm 2\sqrt{ y(0)}, u(0) = - y(0)^2 $$