Proofs in the Appendix A of Sannikov (2007)

Question

I have a few questions about the proofs in Appendix A of Sannikov (2007), Games with Imperfectly Observable Actions in Continuous Time.

1. In lemma 4, when he shows the Lipschitz continuity of $H_a(w,\theta)$ in $\theta$, he derives an auxiliary function $F(\theta^\prime)$, takes its derivative, and bounds that derivative (page 41). How does he get that bound? What is $|\mathcal{V}|$? How is he able to bound the factor involving $\beta^1$ and $\beta^2$?
2. In proposition 4, why does Lipschitz continuity of the objective guarantee continuity of the value function? Does this just follow from the Maximum Theorem? If so, why did we need Lipschitz continuity?
3. Also in proposition 4: why does the initial curvature being positive guarantee that it stays positive?
4. How does idempotency of $Q_i(a)$ guarantee that $\bar{Q} \geq 1$?