How do I construct charts that are used to form an atlas that covers the real projective space ${\mathbb{P}}^n$?

Your question does not deal with charts on $\mathbb P^n$, but with open sets covering $\mathbb P^n$. These sets may be taken as the domains of charts, but that is another question.

The simplest way to answer your question is to observe that the quotient map $\pi : \mathbb R^{n+1} \setminus \{0\} \to \mathbb P^n$ restricts to a quotient map $\phi = \pi \mid_{S^n} : S^n \to \mathbb P^n$. This is easily proved; see also When is the restriction of a quotient map $p : X \to Y$ to a retract of $X$ again a quotient map?

$\phi$ identifies pairs of antipodal points $x, -x$. It is an open map: Let $U \subset S^n$ be open. We have $\phi^{-1}(\phi(U)) = U \cup (-U)$, where $-U = \{-x \mid x \in U\}$. The antipodal map $a : S^n \to S^n,a (x) = -x$, is a homeomorphism, thus $-U = a(U)$ is open. We conclude that $\phi^{-1}(\phi(U))$ is open, and therefore $\phi(U)$ is open because $\phi$ is a quotient map.

For $j = 1,\ldots,n+1$ define $U_j^\pm = \{(x_1,\ldots,x_{n+1}) \in S^n \mid (-1)^{\pm 1} x_j > 0 \}$; these are $2(n+1)$ open sets which cover $S^n$. We have $\phi(U_j^+) = \phi(U_j^-) = \{ [x_1 : \dots : x_{n+1}] \mid x_j \ne 0 \}$ and these sets (which we denote by $U_j$) are open in $\mathbb P^n$. Clearly they cover $\mathbb P^n$. This answers $(1)$ and $(2)$. Note that all $n+1$ of these sets are needed to cover $\mathbb P^n$ because the points $[x_1 : \dots : x_{n+1}]$ with $x_i = 0$ for $i \ne j$ and $x_j = 1$ are contained only in $U_j$ and in no other $U_i$.

Concerning $(3)$: You say "When I learned about homogeneous coordinates only the last coordinate was the one that had to be non-zero to allow division by it". This is not true, the points having this property lie in $U_{n+1}$. But the points $[x_1 : \dots : x_n:0]$ do not lie in $U_{n+1}$. In fact, division by $x_j$ makes sense precisely when $[x_1 : \dots : x_{n+1}] \in U_j$.