Is one sinful who teaches non-Muslims network engineering if they use this knowledge in unlawful activities (such as pubs and banks)? And what is the ruling if the teaching was to Muslims who then worked in banks? And what is the ruling on working for a company that deals with banks, even if indirectly?

It is permissible to teach "networking" to non-Muslims, unless it is known or highly probable that the student will use it for unlawful purposes, or that the people of the country predominantly use it for unlawful purposes. In such cases, it would be spreading corruption and assisting in sin. The rule is not to lease for an unlawful act or that which leads to it, as Allah says: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression). So, if the predominant use of what is sold is for an unlawful benefit, then it is not permissible to sell it. However, if the knowledge is used for unlawful purposes after being acquired, and you are unaware of it or the apparent situation is safe, then the sin is upon the one who uses it.