Interesting question. Your heart rate refers only to the number of times that the heart beats each minute. It does not speak to the quality of your heart's contractions or the effectiveness. So yes, one could have heart failure with a normal heart rate (typically considered between 60-100 beats per minute). Similarly, any heart rhythm that is not sinus (i.e., sinus rhythm, sinus tachycardia, sinus bradycardia) is considered an arrhythmia. So, someone could have a heart rate of 70 bpm per minute (within the normal range), but have atrial fibrillation (which is considered an abnormal rhythm). If you have more specific questions, you should discuss with your doctor.
Yes. But if you have a normal heart rate and the symptoms you have described in your other posts - palpitations, dizziness, occasional shortness of breath,- it doesn't mean you have heart failure. There are lots of benign causes of those symptoms,especially if you are anxious,overweight and unfit. A visit to a doctor would put your mind at rest.