Pareto optimality and Externalities
The idea behind Pareto optimality is efficiency. On the discrete set, there is no allocation where, compared to $(1,1,1,1,1)$, we could make someone better off and keep the others at least indifferent. This is why this allocation is Pareto optimal. However, any other efficient allocation (in the sense that we are not throwing away resources) will be Pareto optimal here as well. 

For the set in $\mathcal R$, we assume a symmetric solution and solve

$$\max_{q\in[0,2]} 5\cdot q(12-(5q))$$
$$\Rightarrow 5(12-5q) -25q$$

The first term is the direct benefit of increasing $q$, the second one is the externality (that we now internalize). The interior solution is given by $q = 1.2$ where everyone gets the payoff $7.2$. We know that the interior solution solves the problem given that the boundaries $0, 2$ both delivered lesser value than $q=1$. 

Finally, for the tax, derive the FOC of the agents given a general tax $c$. Then, look for the optimal quantity you want them to have, that is, replace $q_i$ with $q_i^*$, the welfare-maximizing solution, and then look for the value of $c$ that would solve the FOC (and hence make the farmers optimally chose what you want them to).