Is it permissible for the deceased husband's brother to remain in the widow's home during her iddah, given that she has no mahram among her sons, and is it permissible for her to meet men for condolences?

It is not permissible for a man to remain with a woman if there is no mahram present other than children who do not prevent khalwa (private seclusion), unless the man resides in a separate part of the house. As for a woman offering condolences to men, it is not permissible except for her mahrams, and it is disliked for a young woman to offer condolences.