Learn English – a term for something unwanted but which cannot be ignored

Prerequisite - Co-requisite - Post-requisite -- Where task A is the prerequisite and task B can either be a co-requisite or post-requisite.

As a project manager I have often used predecessor/successor, but usually where there is a chronological order.

Project tasks are related in exactly the same way that some educational courses are. This does imply providing some sort of utility, but not necessarily and the relationship is well understood.

With this in mind, one of my college chem classes was a prerequisite and an associated chem lab was a co-requisite.

I would imagine that whoever told you that (1) is not correct did so based on a prescriptive rule against double negatives. The advice is mirrored here on a list of common grammatical errors in English, and here in an English for Dummies book excerpt. The problem, however, is that it is kind of silly to take something that people say all the time and declare that it's ungrammatical. I don't know of anyone who would hear the sentence "I can't help but think that he's a criminal," and then wonder whether I think he's a criminal just because of the double negative between cannot and but.

As far as I know, (2) is correct, but nowadays it is not usually said or written, except when trying to achieve an overly formal or old-fashioned tone.

For (3), the sentence is correct and is largely interchangeable with (1). This page claims that (1) is primarily used to mean:

I am forced to the conclusion that S[, though I'd rather not believe it].

(or)

I am forced to the experience of S[, though I'd rather not have it].

(where S is some sentence, and the sense conveyed by the braces is optional), while (3) can either take that meaning, or:

I'm always thinking about [the fact that] S[, though I'd rather not do that].

I can see the point, but they both sound fine in either usage to me. Maybe that's just my experience.

One final remark is that all three of the constructions would have a certain formalism to them if spoken; I imagine it is much more common to use the contraction can't in place of cannot in that case.