Am I obligated to visit my sick brother in the hospital, whom I have shunned at my husband's command due to his actions that displease Allah and his common-law marriage, or should I continue shunning him in the hope that he realizes his mistake?

Boycotting a disobedient relative is subject to the شرعية (Islamic legal) interest. If boycotting deters him and reforms him, then it is شرعية (Islamically permissible). However, if maintaining ties with him is more beneficial, then that is preferable. Seek your husband's permission to visit your brother and resume advising him, especially during his illness, for he might be more receptive. If he does not grant you permission, then obey him, and you must supplicate for his guidance. A man marrying an older woman does not violate the شرع (Islamic law). An 'urfi (customary) marriage is valid if it fulfills the conditions of the ولي (guardian), witnesses, and offer and acceptance. However, if it takes place without the guardian's knowledge, it is invalid.