Question regarding General Equilibrium under non-convexities

I have the following question on my problem set:

It's clear to me, since consumer 2 does not care about good 2, that we should give all the economy's endowment of good 2 to consumer 1. In the other hand, both consumers care about good 1.

For item a), I think that in any optimal allocation we should have $x_{2,1} = 1$, since consumer 2 does not care about good 2.

Also, with this in mind, any allocation that has $x_{1,1} + x_{1,2} = 1$ is Pareto optimal, because we can only make one better hurting the other (assuming we have already exhausted good 2, giving all the economy's endowment to consumer 1). Question: is this reasoning right?

Another question: assuming I got it right, I have no idea how to find the vector price for each case.

For item b), the 1st Welfare Theorem needs to hold because there's local non-satiation for both consumers. On the other hand, lexicographic preferences are not convex. So, there's no reason for the 2nd theorem to hold. Is that sound?

Thanks a lot in advance!!

You are right. Set of Pareto efficient allocations consist of all feasible allocations $((x_{11}, x_{21}), (x_{12}, x_{22}))$ satisfying the property that individual 1 consumes all of good 2 i.e. $x_{21} = 1$ and $x_{22} = 0$.
Competitive (or Walrasian) equilibrium in such an economy does not exist. At all price vectors $(p_1, p_2)$ satisfying $p_1 > 0$ and $p_2 > 0$, both the consumers will only demand good 1. As a result we will always have excess demand for good 1 and excess supply of good 2. When we consider a price vector $(p_1, p_2)$ of the form $p_1 > 0$ and $p_2 = 0$, then consumer 1 will demand infinite amount of good 2 leading to excess demand for good 2. Therefore, there does not exist a price vector that clears both the markets.