I keep getting nightmares. Am I depressed? Is something wrong with my mental health?

A nightmare. Nightmare are of many different causes. Trauma or overwhelming life events ala post-traumatic stress disorder can be responsible - psychotherapy specific to that diagnosis would likely work best. Prazosin works well. Medications or alcohol/substance abuse if responsible would be remedied by discontinuing them at least for a time to determine if responsible. Those sensitive may avoid horror movies.
Talk to a therapist. Mental health disorders are defined as something that is interfering with your normal functioning. So unless the nightmares are making you unable to sleep, it's unlikely they would be considered a mental health disorder. However, it may be helpful to you to work out what is going on in your subconscious by talking to a therapist. Psychotherapy can be helpful even if you don't have a mental illness.