Is flexeril (cyclobenzaprine hcl) a drug that can get you kicked out of military service?

No, but... It will ground you if you have airborne or diving status, and your profile will/should indicate you are on muscle relaxants which may impact safety and keep you from doing your MOS. You may be stuck in the motor pool for a while until you are considered safe by your treating military doctor. Being on a limiting T3 (liothyronine) profile long enough - regardless of reason - will send you straight to the MEB.
No. Flexeril (cyclobenzaprine hcl) is a medication that is used at military treatment facilities - the question is if you have a condition that causes impairment.
Probablly Not. Unless you are obvously sedated or impaired. It may not be compatible with certain duties like being a pilot or in other critical situations. You probably risk some sort of discipline for taking a drug which the military is not aware of, however.