For part $a$, I did $P(\gamma_t > x | \delta_t = s) = P(\gamma_t + s > x + s | \delta_t = s) = P(\beta_t > x + s) = 1 - G_t(x + s)$. I am not sure if this is correct since $\beta_t > x + s$ does not imply $\gamma_t > x$ and $\delta_t = s$.
For part $b$, I am not sure. I would like to say something like $P(\gamma_t > x | \delta_{t + x/2} = s) = P(\gamma_{t + x/2} > x/2 | \delta_{t + x/2} = s)$ and then basically use part $a$ again, but the question asks to give the result in terms of the CDF of $\beta_t$, not $\beta_{t + x/2}$ so I am a bit stuck.