Cover for Cech cohomology of the constant sheaf $\mathbb{Z}$ over $\mathbb{P}^n$

I hope that this will not be the only answer to this question, but I'd like the point out that the theorem of Leray gives sufficent conditions for $$H^n(\mathcal{U}, \mathcal{F})\rightarrow H^n(X,\mathcal{F})$$ to be an isomorphism for all $n$, namely that for all $n$

$$H^n(U_{i_0}\cap \cdots \cap U_{i_k},\mathcal{F})=0$$ for all finite intersection of open sets from the cover. This is not satisfied by the open covering given in the question.

$\text{}$1. Is there a standard, or at least well known, cover of $\mathbb{P}^n$ satisfying this condition?

Not an explicit one as far as I know.

$\text{}$2. Is there any intuition on how to choose such covers?

Any covering consisting of geodesically convex sets works.

$\text{}$3. Is there any other direct method of seeing the isomorphism $H^2 (\mathbb{P}^n, \bf{\mathbb{Z}}) \cong \mathbb{Z}$?

The simplest approach in my opinion is to use the cell complex structure of $\mathbb{P}^n$: it has exactly one cell in each even dimension $0$, $2$, $\ldots$ , $2n$. Then use the definition of homology for CW complexes, and apply it to this cell decomposition.