Infinite linear independent family in a finitely generated $A$-module

Well, the key words are onto endomorphisms are isomorphisms. (Actually, the paper proves more than the title says!)

2) The answer is negative and as you remarked this also solves 1). Let $F$ be the submodule of $M$ generated by $x_1,\dots, x_{n+1}$. We have an isomorphism $F\to A^{n+1}$. Since $M$ is generated by $n$ elements there exists $K\le A^n$ such that $A^n/K\to M$ is an isomorphism. Now take the canonical projection $p:A^{n+1}\to A^n$ which sends $(a_1,\dots,a_{n+1})$ to $(a_1,\dots,a_{n})$, and thus we get an onto homomorphism $F\to M$. This must be an isomorphism, and we get that $p$ is injective, a contradiction.

4) Yes, it is! Use similar arguments as above to get that $M\cong A^n$.

3) This follows immediately from Orzech's Theorem.

Remark. A proof of Orzech's result can be found on M.SE in this answer.

(Another proof of these questions, based on McCoy's Theorem, can be found in this paper.)