Question regarding General Equilibrium under non-convexities
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 $p_1 &gt; 0$ and $p_2 &gt; 0$, 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 $p_1 &gt; 0$ 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.

First Welfare Theorem holds because preferences satisfy Local Non Satiation. Second Welfare Theorem does not hold because no matter how we distribute endowments, competitive equilibrium in such an economy does not exist, but efficient allocations exist.

Also, please note that the lexicographic preferences are convex.