What is the ruling on a salary, and to what extent is it unlawful or lawful if a person swore not to deal with banks and then dealt with them? And is he only sinful if he does not offer expiation, or does the salary become unlawful?

We advise you to turn away from the whispers of doubt regarding oaths. The basic principle is that you have not sworn an oath as long as you are in doubt, because the default state is the blamelessness of one's conscience. Even if you assume that you swore an oath, broke it, and did not offer expiation, this does not affect your salary, as there is no connection between the two matters.