@IgorRivin Thank you for your kind reply. I apologize for bringing up the unpleasant experience in a comment when the post was by none other than the paper's co-author. It was inappropriate.

@IgorRivin Ah, I'm sorry if my careless comment came out sounding sarcastic by bringing up that particular example about your paper; I should've checked who wrote the answer before posting the comment... Also, thank you for the interesting tidbit!

This is what I hear from senior mathematicians in my field (combinatorics). I wonder if things are slightly different between fields of math, regions (e.g., West vs. Asia), generations, etc. If mathematics had a universal culture about the order of authors with no exception, Inventiones wouldn't have had to publish a corrigendum like this one...: link.springer.com/article/10.1007%2FBF01232247 (To give credit where credit is due, I found this example in this MO answer mathoverflow.net/a/20085/27829)

This map of journals in all fields of science may interest you: vosviewer.com/maps If I understand correctly, it's visualized in a way the more citations between given two journals have, the closer they're on the map. I don't know how much citation directions are considered though. In any case, "prestigious mainstream pure math journals" form a clique-ish circle on the edge of the map. This might give a rough, pseudo-objective measure of how far discrete math is from the mainstream math circle in a way you described (except maybe less use of the "use" vs. "give back" information).

I don't think so unless I'm missing something trivial. The former question requires you to list up all possible nonisomorphic ones for each $v$, which is quite a difficult problem even for a fixed $v$ if it's large. In fact, it's already very difficult to enumerate all of them (i.e., compute the number of nonisomorphic ones) for fixed $v$ and $n$, let alone to construct all of them...

As for your questions, the answer to the first one seems very likely to be "no" because we need to know all inequivalent cyclic difference sets that belong to $C_v$ for infinitely many $v$. You can check all relevant papers published in these several years; older results should be covered by the linked article, CRC Handbook, and reference therein. In principle, you should be able to tell if the answer to the latter question is known on your own this way as well, although it might require a lot of reading... I'm guessing it's not known yet if such $N$ exists.

A typical definition of equivalence between $D_0$ over group $G_0$ and $D_1$ over $G_1$ is that they're equivalent if there is a group isomorphism $\pi$ between $G_0$ and $G_1$ such that $\{\pi(d) \mid d \in D_0\} = D_1 +g$ for some $g$. In your case, $G_0$ and $G_1$ are both cyclic. So, we only need to consider the group automorphisms, i.e., the unit group. So your definition of equivalence is pretty much the same. Equivalent difference sets are all isomorphic. But the converse may not be true, although in the cyclic case, as far as I know, all known isomorphic difference sets are equivalent.

A common definition (given in CRC Handbook of Combinatorial Designs) is that two difference sets $D_0$ and $D_1$ are isomorphic if the designs $\operatorname{dev}(D_0)$ and $\operatorname{dev}(D_1)$ are isomorphic. As you probably already know, by developing a difference set, you get a sharply point-transitive design. So, if the corresponding designs are isomorphic in the usual sense in design theory, your difference sets are isomorphic. This notion is different from the equivalence you wrote there. (cont.)

I guess you identify two isomorphic cyclic difference sets when defining the set $C_v$...? If that's the case, are $p$ and $f$ fixed, too? For example, is $\vert C_v \vert$ the number of all nonisomorphic ones of the same parameters? Or do you mean $C_v = \bigcup_{p, f} C_{v,p,f}$, where $C_{v,p,f}$ is the set of nonisomorphic difference sets of order $v$ with $k-\lambda = p^f$ and $\operatorname{gcd}(k-\lambda, v)=1$? In any case, when you say, "$C_v$ is explicitly known," do you mean a complete characterization of cyclic difference sets (up to isomorphism, I guess) is known for some cases?