Are my siblings and I considered children of adultery because of my father's marriage to my maternal aunt and his admission to a mental hospital, with doubt concerning his divorce from her before his marriage to my mother and our birth?

It is unanimously forbidden to marry two sisters simultaneously, based on Allah's words: "And that you take [in marriage] two sisters simultaneously." If the father marries the mother after the aunt's divorce and the expiry of her 'iddah, there is no issue. However, if he marries her before that, the marriage is invalid, and he must separate from her. Then, he is permitted to marry her with a new contract after her 'iddah expires.

Even though the marriage is invalid, lineage is established due to the semblance of a contract, and the children are not considered illegitimate. Ibn Qudamah said: "Lineage is attached to him; for it is either from a valid marriage or an invalid marriage, and in both, lineage is attached." Sheikh Ibn 'Uthaymeen was asked about a man who married a woman and her niece, and he replied: "Their marriage is not valid; rather, it is void, and it is obligatory for him to separate from this last wife... However, if she bears children from him while ignorant [of the prohibition]: then the lineage of these children is attached to their father, so they are children of their father and mother."