As an auxiliary verb, have has two distinct meanings (well, more—but two that are relevant here):
With a passive participle, it means:
With an infinitive, it means:
- to see to it that someone does something (“I’ll have my secretary send you the documents”).
In the sentences you mention, the meaning is supposed to be the first of these two, and they therefore require the passive participle, rather than the infinitive. Using the infinitive is not grammatically wrong as such, but it does force the sentence into the second meaning, which makes it semantically quite strange:
The rope had its surface wear.
The rope had its surface get worn.
These two sentences are both grammatically fine, and they make sense—but they are not likely to ever be spoken by anyone, because they mean the following:
The rope made/ordered its surface [to] wear [itself].
The rope made/ordered its surface [to] be worn by something else.
These are both extremely unlikely situation in this universe.
The rope had its surface worn.
– is better, since it only states that someone or something had worn down the surface of the rope. But it is still a bit awkward, since, even in the sense of suffer something being done, this construction still implies a certain amount of ability to act (or at least feel). So for simple inanimate objects, it is more common to use a simple passive or adjective instead: “The surface of the rope had been worn/was worn”.
In your longer sentence, the semantics become even more convoluted.
The wire is passed through the pliers in a specific way to avoid having it get bent.
This arguably means, if expanded to more literal, descriptive constructions:
You pass the wire through the pliers in a specific way in order to make sure you do not make/order the wire [to] be bent by something (else).
That is a mouthful. Far too many modals for one sentence. If you start telling people to do something in order to make sure that they don’t see to it that a thing gets [verbed] by something, you’ll have a very lost and confused audience on your hands. It would be much more natural to simply say:
The wire is passed through the pliers in a specific way to avoid bending it.