Is an apartment, whose down payment was made by the wife's mother from money she acquired from interest from a non-Islamic bank, considered unlawful for the spouses, and if so, must the unlawful amount be identified and given away in charity?

Usurious gains are forbidden. Your mother-in-law must repent to Allah and dispose of these gains by spending them on general Muslim welfare or giving them to the poor and needy. There is no sin upon you or your husband for benefiting from the apartment she purchased with her mixed funds, for the sin of usury pertains to her own responsibility, not to the money itself. You are not required to estimate the unlawful money and give it away in charity.