Answer to Question 1
Adverse selection is the situation in which one party to a transaction takes advantage of knowing more than the other party to the transaction. Moral hazard refers to the actions people take after they enter into a transaction that make the other party to the transaction worse off. An important difference between the two is that adverse selection refers to what happens at the time of entering into a transaction (for example, a reckless driver buys automobile insurance) while moral hazard refers to what happens after entering into the transaction (for example, the insured driver drives his car off a highway and into a tree).
Answer to Question 2
To show that both countries are better off it is necessary to demonstrate that total consumption, not just production, of both goods is greater after trade. If a country specializes completely in producing one good apples, for example it has given up the opportunity to produce another good that consumers value; let's say this other good is plums. Apple lovers now have more of the good they like, but the country as a whole cannot be better off unless the change in production benefits plum lovers too. This can be done by trading some of the additional apples that are produced for some of the plums the other country has produced. Trade may benefit apple lovers more than plum lovers (or vice versa) but if, after trade, more of both goods can be consumed then trade has unambiguously made all consumers better off than they were previously.