Not really. You could have a form of periodontal disease. The bacteria that has progressed under your gums, causes a loss of attachment from the teeth and usually "pockets" form that the dentist will evaluate. And in some, they will notice that the gums feel "loose.".
No. You may have very inflamed gums due to active gum disease. See a dentist for a definitive diagnosis.
Not normal. You may have periodontal problems. See a dentist at once for evaluation and possible treatment of the problem.