Short answers, pulled from the story.
Pareto efficiency is named after Vilfredo Pareto (1848-1923), an Italian civil engineer and economist who developed the concept while studying economic efficiency and income distribution. Pareto originally used the word "optimal" for the concept, though "efficiency" is considered a more accurate label because the concept identifies a set of possible outcomes rather than a single best one.
An allocation is Pareto efficient when it is impossible to make any one party better off without making at least one other party worse off. All possible Pareto improvements, meaning changes that benefit someone without harming anyone, have already been made.
Economists Kenneth Arrow and Gerard Debreu were the first to prove mathematically that a competitive market leads to a Pareto-efficient outcome under the assumptions of the first welfare theorem. Those assumptions include perfect information, no externalities, perfectly competitive markets, and the existence of markets for all possible goods.
A strong Pareto improvement makes all agents strictly better off, while a standard Pareto improvement requires only one agent to be strictly better off with no one made worse off. Weak Pareto efficiency means no strong Pareto improvement is possible; strong Pareto efficiency means no weak Pareto improvement is possible either.
No. Pareto efficiency does not require an equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. As Amartya Sen's liberal paradox shows, the goal of Pareto efficiency can also conflict directly with the goal of individual liberty.
Researchers studying bacteria found that genes tend to be either resource-efficient (inexpensive to produce) or translation-efficient (easier to read). Natural selection pushes highly expressed genes toward the Pareto frontier for these two competing demands, and genes near that frontier also evolve more slowly, indicating a selective advantage.