Does the husband bear a burden of sin for what happened between his wife and his brother, and does he have the right to search into a past that is gone, especially since he has nothing to arouse his suspicions?

Your husband has fallen into what the Prophet, peace and blessings be upon him, warned against: men entering upon women, especially the in-law (the husband's brother and his relatives), because their danger is more severe and could lead to adultery. It is not permissible for your husband to search into the past after your repentance; rather, he must conceal it, as this will corrupt his heart. The Prophet, peace and blessings be upon him, said: "Avoid these abominations that Allah has forbidden. Whoever commits any of them should conceal himself with the concealment of Allah and repent to Allah." It is commendable for one who has committed a sin to repent and conceal oneself, and not to inform anyone about it. And whoever becomes aware of it must conceal it and not expose them. A man has no right to search into the past about which his wife has repented, and a woman should not tell her husband what happened and from which she has repented.