I got nose surgery done and one of my dissolving stitches came undone.

You should talk to your surgeon. I do not allow my patients to place a q-tip or anything else in their nose for at least one week for just that reason and the risk if bleeding.

Not being able to breathe through the nose immediately after rhinoplasty is quite common due to crusting and swelling inside the nose. I recommend avoiding putting anything in your nose but likely a stitch coming out inside your nose is of little consequence. A stitch on the outside of the nose may be of greater consequence. Please consult your doctor.