Indeed, the case when $A$ has nilpotent elements is essential to the theory! Check out Grothendieck's article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes...": scholar.google.com/scholar_url?url=http://cm2vivi2002.free.fr/… Essentially, degree $d$ differential operators on $A$ are certain endomorphisms of $A \otimes A/I^d$, where $I$ is the augmentation ideal.

If you care mostly about sheaf cohomology (as opposed to the general theory of derived functors, which I'd argue is essentially the same thing as the general theory of injective resolutions etc or of derived categories), then Cech cohomology is pretty concrete and conceptually simple. (The fact that these have long exact sequences is then a theorem rather than a definition).

Do you mean to have the coefficients $\mathbb Q_p$ and the base field be the same? The Weil conjectures are about cohomology with coefficients in $\mathbb Q_\ell$ for some $\ell \neq p$, and do indeed have a good generalization to non-smooth (and non-proper) varieties. This is Deligne's "Weil II" result, and is related to the size of Frobenius eigenvalues. When $\ell=p$, the study of $p$-adic Galois representations arising as etale cohomology is the subject of $p$-adic Hodge theory, and the central theorems carry over to the non-smooth case there too.

The definition is given above, but yes, this is power series whose terms go to $0$. You can also think of it as the completion of $\mathbf{Q}_p[T]$ in the $p$-adic topology.

I should add that the generic fiber functor commutes with fiber product, so if $\mathfrak{A}, \mathfrak{B}, \mathfrak{C}$ are formal models for tigid-analytic spaces $A,B,C$, we have $(\mathfrak{A} \times_{\mathfrak{B}} \mathfrak{C})_K =A \times_B C$

Usually fppf suffices, and it's easier to work with - sheafifications exist, fppf maps are open, etc. However, schemes and algebraic spaces do satisfy fpqc descent (the latter is a non-trivial theorem of Gabber). One place I've seen the fpqc topology appear is when working in the category of perfect schemes, where maps are rarely finitely presented. Then again, it seems that in this setting, more exotic topologies like the v-topology or arc-topology might be preferable. (See arxiv.org/abs/1407.8519, arxiv.org/abs/1507.06490, arxiv.org/pdf/1807.04725.pdf)