“When should I be concerned about a bloodshot eye?”

Assuming you don't have a bleeding disorder or uncontrollable hypertension, not too much. Often, these are the result of straining, Valsalva, or coughing hard. If you have a blood disorder that causes excessive bleeding, or you have uncontrolled hypertension, see your primary care doctor to confirm that those conditions are not the problem.

Depends on the diagnosis. A complete ophthalmologic exam should be performed.