Optimal Price and Quantity of Externality
Your steps look correct. There is one small typo in the first $\frac{\partial \mathcal L}{\partial h}$: the term $-4h(h-10)$ should have been $-4(h-10)$.
The results also look reasonable: Regardless of who makes the offer, the Pareto optimal level of $h$ is produced. Given how the bargaining procedures are structured, whoever gets to make the offer gets to keep the surplus of $20$.

P.S. There is an issue in part 2 of the question. It says: &quot;If the firm rejects the offer it cannot produce so the firm's profit is 0.&quot; This statement is a contradiction. If the firm cannot produce ($h=0$), then profit would be $-80&lt;0$; if its profit is $0$, then it must produce some positive amount of $h$. But this lack of rigor is on whoever wrote the question, not you.