If we were contracting "I do not need to go" and "I do not want to go", we could indeed say "I do not need nor want to go."
Here, however, due to the nature of the verbs involved, the two statements we want to combine do not share the same construction: "I cannot go" and "I do not want to go". In negative statements, ordinary verbs require do-support (as in "do not want"), but this construction would not work with modal auxiliaries such as "can". Therefore, "I cannot nor want to go" feels awkward, because the second verb "want" is lacking the expected do-support. The somewhat less-elliptical "I cannot go, nor do I want to", on the other hand, fulfills this expectation and therefore feels proper.
Seems okay to me. I guess the more common thing to do would be to shorten the cannot to can't, but other than that I don't see anything wrong with it.
Could we also say "eu não posso ir, nem quero" or does the infinitive have to be at the end in this construction?
Both work. It's similar to English, you could say "... nor do I want to" or "... nor do I want to go" - I guess also in both languages it's more common to omit the "go" (Eu não posso ir, e nem quero) C: