1.- He picked up neither of them becAuse they wERE extremely expensive. //He will pick up either of them since they are the same. (Pronoun) Both good
2.- I neither know Japanese nor Cantonese // I can cook either pasta or rice, what would you like? (Conjugation) Both good
3.- Neither job suitS me at all //Either shoes might like him, choose one. (AdjeCtive) second one makes no sense
4.- I wouldn't use the car without his-her permission, I would use his-her computer, neither. No
Can you tell me if I've understood well or If I've misunderstood, please explain me how I should use them.
If working with the negative, that then you use neither or neither nor. If working with the positive you use either or either or. You don't mix either nor or neither or. There is a lazy way to say drop neither at the end of a sentance, don't.
4.- I wouldn't use the car without his-her permission nor would I use his-her computer (actually his-her is just wrong but move on)
3.....He might like either shoe, choose one