For 1, spaces in which the discrete subsets are at most countable, and these are called spaces with "countable spread" in topology. Here, the spread $s(X)$ of a space $X$ is defined as the supremum of the cardinalities of all discrete subspaces of $X$, where by convention a finite supremum is rounded up to $\aleph_0$ (only infinite cardinals are used), also because every infinite Hausdorff space has a countable discrete subset (so spaces with a finite spread would be "pathological" non-Hausdorff spaces, or finite to begin with).
If a space is second countable, then every subspace is second countable too, and a discrete second countable space is at most countable, so a second countable space has countable spread. But this argument can be repeated for other classes of spaces: if every subspace of $X$ is separable ($X$ is then called hereditarily separable) or every subspace of $X$ is Lindelöf ($X$ is then called hereditarily Lindelöf) then $X$ has countable spread too (as a Lindelöf discrete space or separable discrete space both must be countable). For metrizable spaces, countable spread is equivalent to being separable, or Lindelöf, or second countable. See my post on topology atlas, but in general this need not be the case. But the Wolfram quote maybe comes from the fact that a lot of mathematics is done in separable metrizable spaces, like the Euclidean spaces.
An example of a separable compact space that does not have countable spread is $\beta(\omega)$ or $[0,1]^{\omega_1}$.
As to 2, the property that all finite subsets are discrete is equivalent to being $T_1$ (defined either as all singleton sets are closed, or for every $x \neq y$ in $X$, there are open sets $U$ and $V$ such that $x \in U, y \notin U$ and $y \in V, x \notin V$). This already follows from considering subsets of 2 points.
As to 3, adding a discrete topology to a set doesn't make it any more topological, as all functions on it are continuous, there are no non-trivial convergent seuqneces or nets, etc. So a discrete topology adds no information. It's true, for example, that any group can always be given a discrete topology and then it's a topological group (the group operations are continuous), but if we apply theorems from the general theory of topological groups, we cannot prove anything new that we couldn't prove by just plain algebra/group theory. The same holds for other types of (finite or not) structures in discrete mathematics: discrete here is opposite to "continuous", one could say: we do not consider topological or analytical structure, but just the structure as a set. The discrete topology is as informative as no topology in this case....