Very unlikely. Tetracycline (TCN) is an antibiotic, and is not known to have withdrawal symptoms. A rash just after using TCN for 3 months is unlikely to be a withdrawal symptom. However, the rash could be due to an allergy to TCN, even though TCN was stopped. Persons allergic to TCN may also be allergic to Doxycycline & minocycline. Also, the rash might be just a coincidence and have nothing to do with TCN.
No. There is no known withdrawal syndrome associated with tetracycline. You may have had a rash that the tetracycline was suppressing, however, such as rosacea or acne.