Relationhips between successorship, enumerable, and countable

You write:

"If each element in a set has a successor, then the set is enumerable."

That's false: the existence of a successor operation says nothing about cardinality (besides, of course, the fact that any linear order in which every element has a successor must be infinite).

Intuitively, the existence of successors is a local condition. Knowing that in a linear order $\mathcal{X}=(X,\triangleleft)$, for each $x\in X$ there is a unique $y\in X$ such that $x\triangleleft y$ and no $z$ has $x\triangleleft z\triangleleft x$ doesn't do anything to control the size of the whole linear order $\mathcal{X}$ itself. In fact there is a precise theorem behind this intuition: the Lowenheim-Skolem theorem. No "simple structural" (= first-order expressible) property can distinguish between countable and uncountable infinities.

• Pet peeve time! Lowenheim-Skolem is really two distinct theorems. The "upwards" part is just a corollary of compactness, while the "downwards" part is a genuinely different result. Lumping them together as "the Lowenheim-Skolem theorem" is pedagogically suboptimal in my opinion: it suggests that the two parts should be true for the same reason, but they really aren't. (FWIW in the context of the OP, both parts are relevant but the "upwards" part is a bit more relevant.)

Now it is true that if $(X,\triangleleft)$ is a linear order with successors such that every element of which can be reached from some single element by applying the successor operation finitely many times, then $X$ must be countable. However, this is a vastly stricter condition than merely having successors exist at all.

Meanwhile, "enumerable" and "countable" are synonymous (OK some texts do introduce a distinction between the two by having one allow finiteness and the other not, but this isn't relevant here), so there isn't much to say there.