Is money earned from training at an Islamic bank branch, which is affiliated with a commercial bank, permissible or impermissible, especially if some operations inconsistent with the "profit and loss" principle are observed?

You are not accountable for the money you took if your work and training were in an Islamic branch of a commercial bank, and you were not certain of the impermissibility of its transactions, especially if the branch had Sharia (definition) oversight or the bank was in the process of converting to an Islamic one, and you did not personally engage in any forbidden act. However, if it becomes clear to you that the branch conducts interest-based transactions, then dispose of the money you possess by giving it to the poor and needy or spending it on charitable causes.