The trauma from loss of a pet has been known to result in; not only Depression, but PTSD (Post-Traumatic Stress Disorder), in which the individual begins to avoid having pets or dealing with pets, and actually experiences anxiety attacks whenever they run into pets that remind them of their deceased pet.
You will of course, need to have a full Psychiatric evaluation asap; to see whether or not your symptoms warrant a diagnosis and treatment for Depression; which may include Psychotherapy and Medication Management, depending on the severity of your symptoms.
On the other hand; any delays in receiving treatment can result in much worse scenarios, including your inability to complete high school, and complete interference with your entire life; as a result of a very poor or mediocre academic and social functioning, or even suicide.
Please inform your parents immediately; who have the legal consent for your medical treatment, that you have been strongly advised by a Board Certified Child Psychiatrist to obtain treatment asap PLEASE!!
As such, how you feel ; determines whether you have any mental illness in the first place. So, the short answer is; being evaluated by a Mental Health Clinician, preferably a Board Certified Psychiatrist; and if you are below 18 years of age, a Board Certified Child and Adolescent Psychiatrist; is the only way to categorize the signs that you are referring to.
That having been said, if you have any strange feeling about yourself or your environment around you, or feel sad and tearful most of the time, or have to drink alcohol or use cigarettes or drugs to be calm or to sleep, or feel paranoid or suspicious of the motive of others (without justification), or hear voices or see things, or have anger problems-to mention a few; then you definitely need to talk to an age appropriate Board Certified Psychiatrist, and he or she will tell you if you have any signs of mental illness or not.