Blaise, I feel so sorry that this situation has put you in the middle. Fourteen is a rough age as it is without all of this adult stuff interfering.
First of all, as a mother, let me just say that the best way to keep your Mom from threatening you when you don't do chores is...you know the answer. Do your chores.
However, with that being said, let me ask...do you know if there is a court order awarding your father visitation? If there is, your Mom cannot withhold visitation from your father when he is supposed to have you without being in contempt of court. The problem here is it sounds from what you say like there is an order but it doesn't specify exact days/times your Dad is supposed to have you. Is that correct? If so, your Mom is pretty "free" to keep you from your Dad because visitation is too much under her control. The first thing your Dad needs to do is petition to have the order modified and specific days/times set. Then he has something to work with and your Mom will know EXACTLY when you are supposed to be with your Dad and your Dad will be able to file contempt of court charges if your Monther doesn't allow you to visit when you are supposed to. Keep in mind, if you get a lot of visitation now, this may backfire as specifics usually mean an evening or two during the week, every other weekend, alternating holidays, some school breaks, etc. Your Dad's attorney can advise him on what he can expect in your state.
It sounds like you Dad has already filed a petition for a change of custody, which is good, but please understand...custody doesn't change for no reason. If you have lived with your Mom all your life it is going to be very difficult for a judge to justify a change. From all the things you say, your Mom's family seems to have some "issues," but not your Mom specifically. This isn't about her family's fitness to parent. It's about her. Your father is going to have to prove your Mother isn't fit to parent and that isn't likely to just happen without solid proof that your Mother is doing something terrible like taking drugs, committing serious crimes, or abusing or neglecting you. Withholding visitation will certainly not bode well for your Mother, but that is harder to prove without a specific court order. You'll have to tell us more about how your Mom is "out there" in order for us to determine whether or not that will have any bearing on the custody case. More than likely, her sexual orientation will have no bearing, unless she has sex in front of you, which obviously isn't occurring since you aren't even positive she's a lesbian. And making you do "chores" or other reasonable things you may not like - like go to school every day, go to church on Sunday, eat your vegetables, maintain good grades, go to bed at a reasonable hour, abstain from alcohol, drugs, and cigarettes, etc. are only going to look good for your Mom. That's what she's supposed to do - keep you safe and help you grow into a solid citizen.
The best thing your Dad can do is document EVERYTHING. EVERY TIME he is denied visitation. EVERY TIME you are threatened by your Mother or accused by her family - EVERY little bit will help.
Also, your Dad should petition to have the judge speak with you privately in chambers so you can corroborate the things he says ("Yes, my Mom threatens me;" Yes, my grandpa calls me a devil worshipper," etc.), and you can make your wishes to live with your Dad known. Mind you, children are NOT permitted to decide whom they live with - the judge will make that decision, but at your age, your voice can be heard.
Also, it is expensive, but well worth it for your Dad to ask for a custody evaluation and a Guardian ad Litem. Having just gone through a nasty custody battle over my husband's 12-year-old daughter (and winning, by the way), I know how valuable BOTH of these things can be. Let me explain what they are.
A Guardian ad Litem is the child's advocate. This is sort of like your lawyer. He/she represents you in court and is charged with looking out for your interests. The GAL will interview everyone involved, including those nutty relatives if you ask him/her to do so, and will file a recommendation with the court as to whom he/she believes should have custody.
A custody evaluator is a psychologist who will also interview everyone, do psychological profiles on your parents and probably your stepmom, and will also make a recommendation with the court.
These two people will hold alot of weight with the court. The decision the court makes will probably depend heavily on their recommendations.
In the meantime, it may be a good idea for you to talk to your school counselor. It will give you someone to talk to and help you sort the whole thing out. It will also help to repair and/or strengthen your relationship with your Mother AND your Dad.
You are going through a very tough ordeal. Stay strong. You (or your Dad) can contact me via the e-mail feature on this board if you need any help, support, information, etc. I am not an attorney, nor am I in LA, but I can certainly share my recent experiences, which are very similar to yours.
Good luck to you. I will keep you in my prayers.