Repair. At 45 years old, you should absolutely seek out a shoulder specialist and have your rotator cuff repaired. The long term risk with leaving it unrepaired is substantial. You are likely to eventually get worsening function and secondary arthritis if you have an unrepaired cuff tear for more than a decade. If you were 75, not such a concern, but at your age, this should not be ignored.
It depends. If the patient has a full thickness tear and is symptomatic then surgery is the appropriate treatment. If a patient does not have symptoms then surgery is at the discretion of the patient and their physician. I tend to tell patients that their are consequences that are irreversible with regard to muscle atrophy if the tendon is left torn. This can lead to more problems later in life. Thanks.