Is it permissible for an Islamic bank to spend from the account of illicit gains on advertising and promotion, or training and research, and what must it do if the impermissibility of that is proven?

It is not permissible for the bank to benefit from funds acquired through illicit means, nor to spend them in ways that bring it profit. Rather, it must dispose of them by spending them on public welfare for Muslims, such as sponsoring orphans and building hospitals.