"This property is local on" : properties of morphisms of $S$-schemes

Ok, this will be a long one.

There is one fuzzy question underlying all your questions : for a given topology (the "classic one" as you say for, but also for Grothendieck's ones, like étale, smooth, fppf, fpqc, synthomic, Niesnievitch etc) what are reasonable properties that one can reasonably expect from a class/type of morphisms of schemes ?

For the classic topology, essentially all classes of morphisms have the following properties : closed under composition, base change and they are local on the base, or target - or on $Y$ with your (2)'s notation. For (1) one says local on the source.

The latter means that to ensure that a morphism $f : X \rightarrow Y$ is in the class, you need only to check that corestrictions on an open cover of $Y$ are in the class. You wrote all covers, but usually one suffices, as practically the properties of the class are often nice enough to imply that if you have this for one cover you have it for all. Same remark for (1), (3) and (4), with all replaced by one. Remark : the construction of product $X\times_S Y$ in the category of schemes over a fixed scheme $S$ is local on $X$, $Y$, $S$. See EGA I 3. 3.2 I think. There won't be a definition of (4), but you will feel what it means.

By the way, the properties of the considered class of scheme are often nice enough to ensure that you can replace in what I previously wrote "open covers" by "open affine covers". This is essentialy dued to the fact that reasonable classes of morphism are defined by properties Q of morphisms themselves defined via properties P of schemes defined in the following way : the morphism $f : X\rightarrow Y$ has the property Q if (and only if but it's a definition) for every open affine $V\in Y$ the scheme $f^{-1} (U)$ as the property Q.

On note of affine covers and checking something on one or all over, one question could be asked : if you check some property of morphism via an affine cover and if I check it on another affine cover, what can we say ? That is, what does what I did on my cover implies on yours, etc... For this matter, there are two key facts :

(1) if you take $U,V$ open affine subschemes of a scheme $X$, then $U\cap V$ is union of open subsets of $U$ and $V$ simultanously (obviously) that are distinguished in $U$ and $V$. (Remember that an open $V \subseteq \textrm{Spec}(A)$ is distinguished if $V = D(f)$ for some $f\in A$.

(2) Let P be a property that affine open subset of a scheme $X$ may have such that :

(a) If an open affine subset $U \simeq \textrm{Spec}(A)$ of $X$ has the property P, then for any $f\in A$ the distinguished affine subset $V \simeq \textrm{Spec}(A_f)$ of $X$ has also the property P

(b) If $f_1,\ldots,f_d \in A$ generate $A$ and if for each $i\in\{1,\ldots,d\}$ the open affine subset $V_i \simeq \textrm{Spec}(A_{f_i})$ of $X$ has the property P then the open affine subset $U \simeq \textrm{Spec}(A)$ of $X$ has the property P

Then if $X = \cup_{i\in I} U_i$ where each open affine subset $U_i \simeq \textrm{Spec}(A_i)$ of $X$ has the property (P), then every open affine subset of $X$ has the property.

Example of properties local on the target : quasi-compact, finite type, open immersion, closed immersion, immersion, finite, quasi-finite, etc

Example of properties local on the base and on the target : locally of finite type, locally of finite presentation, flat, étale, unramified, smooth, etc

For your question on properties of morphism of rings that are local on various spectra, you may guess that these properties often "translate" to the associated morphisms of affine scheme, and that various localness can be checked thanks to what we have discussed previously. There are plenty of nice of fun exemples of this in *Anneaux locaux henséliens" from Michel Raynaud, LNM, 1st chapter for instance. For sure in other books also. (SGA 1 also...)

Finally, how what preceeds could be translated to the étale topology for instance, or others Grothendieck topologies ? This is intimately linked to descent, but I will give only a simple definition : let $\mathscr{T}$ be a Grothendieck topology from the following list : Zariski, fpqc, fppf, syntomic, smooth, étale. Let P be a property of schemes over some base $S$. P is said to be local on the base for the topology $\mathscr{T}$ if the following is true : for any $S$-morphism $f : X \rightarrow Y$ and any $\mathscr{T}$-covering $\{Y_i\rightarrow Y\}$ of $Y$, the morphism $f$ has the property P if and only if for each $i$ the morphism $Y_i \times_Y X \rightarrow Y_i$ has the property P.

Note that as isomrphisms are always $\mathscr{T}$-coverings, any property local on the base for $\mathscr{T}$ is preserved by base change. You may understand why base is important, and maybe why I required it earlier in reasonable properties of a class of morphisms of schemes. The reason is the following : if $V$ is (classic) open in $Y$, the open $f^{-1} (V)$ is but isomorphic to $V \times_Y X$ and $f : f^{-1} (V) \rightarrow V$ is none but $f : V \times_Y X \rightarrow V$, base change of $f$ by the open immersion $V\rightarrow Y$. You see that in fact one mimics classic topology (rejoining earlier definitions) in the setting of Grothendieck topologies, thanks to categoric constructions. Finally, how to relate definitions in the classic topology case to the Zariski topology. They are identical, by definition of a Zariski covering (a family of open immersions with covering images) and by the previous categoric remark.

I am almost off-topic so I will stop here, saying that SGA IV 1. first exposés or Milne's Étale Cohomology are good places to learn about all of this.