Should an eighty-seven year old alzheimer's patient have thoracic surgery to clear a blocked carotid artery?

Only if symptomatic. Age or pre-existing conditions, outside of terminal disease, should not be a contraindication for intervention. Medical management is the primary treatment for carotid disease, followed by medical and interventional therapy. In an 87 year old, surgery would be preferred if indicated.
No. Surgery for a blocked carotid artery is done in the neck, not with thoracic surgery, which is done in the chest.
Let me ask you. There are two questions you need to be addressed. #1 what is the rationale for ordering test or having procedure(s)? #2 what is the quality of life? So to answer that question, how long does he/she suffer alzheimer's? Does this person able to cooperate during this difficult procedure, even before and after surgery? The average life expentency for alzheimer's disease is about 8 years.