“Can you breathe on your own under general anesthesia?”

Yes, you can absolutely breathe on your own under general anesthesia. The only thing that should prevent that is either administration of a lot of opiates, which will the pressure drive to breathe, or the usage of a paralytic, which will take your ability to use muscles that you normally breathe with. Generally speaking, paralytics are only used in cases that require them. Abdominal wall cases, or cases where movement would be a major issue like working on your brain or your eyeball.

Some anesthesia techniques allow you to breathe spontaneously, but the majority control ventilation for you.

Yes, there are certain methods where spontaneous breathing is maintained.