Is the money inherited by the mother from the sale of her father's house, and then traded with by the sons with her knowledge, considered the property of the sons, and do the children of the deceased son have a share in it?

If the mother did not declare the money as a gift, then it is her money and you must pay it back to her.

As for the profit: - If the money was a loan from her, then the profit is for both of you. - If it was not a loan, but for trade by way of donation, then the profit is hers. - If you were not donors, then the profit is divided according to the Mudarabah agreement. - If you did not agree, then the Mudarabah is invalid (fasidah), and all the profit belongs to the mother, and you are entitled to a fair wage (ujrat al-mithl), because an invalid Mudarabah makes the profit for the owner of the capital, and the worker gets a fair wage.

Your brother's share of the profit or fair wage is distributed to his heirs after his death.