I would think your chances are very slim that you have a malignancy. But, you are correct in that dermoids can be malignant. They say 1-3% have a malignant transformation. It is certainly not the norm though.
It sounds like you have a hormonal imbalance due to the changes in your cycle. Other types of cysts can form when hormones are out of sync. I don't know your age, but could you be in perimenopause? It can cause cycle changes. With your missing ovary, you may have been thrown into it a bit sooner than normal.
It is terrible to be without health insurance and need care. You can contact your local health department and try to find a low or no cost clinic. A transvaginal ultrasound can determine if you have a cyst.
I am sorry to hear about your father. It truly is very unlikely that you have a malignancy. Most cysts are benign and since you have a benign pathology from your past cyst, it is less likely that a malignancy has occurred this soon. But, with your symptoms, you really should see a doctor to be sure all is well.
I don't think there is any relation between dermoids and cancer, so don't get too stressed out about that. I assume the dermoid was confirmed at a pathology lab right?
Do get those irregular periods checked out though... if you don't have insurance, try to see a gyn and have an ultrasound done. It might cost you a few hundred dollars...?