Okay.

Step one: Relax. Breathe.

It's difficult for people to think clearly when in an emotional state (this goes for you, and the people you live with). I'm not saying what they do, or what you do is right or wrong, but this situation has clearly escalated to levels it should've have to begin with.

Here's a few things I've picked up from what you just said (granted, I do not live with you, know you, or claim to know you, but I do know households with distress and angst. And there are tons of ways to work around the negative situations.)

They were right to tell you that they don't have to call you whatever gender pronoun it is that you desire -- just as you are right to not agree with them and/or their political/religious views. There is literally NO reason for someone to be thrown out on the streets for a conflict that can be resolved through communication.

Is venting personal/family issues on Facebook a good idea? Probably not. But we learn from our mistakes. This could be seen on their side as immature, and breaking their trust. To be frank, I wouldn't want strangers to know what goes on inside my household, and/or the relationships I have with my family. The Facebook thing was more than likely a "heat of the moment" thing. Whatever, they will get past it.

Their daughter, my cousin, thinks that I'm being unrealistic and unreasonable.

I mean, are you? Do you put yourself in their shoes? Sometimes I don't think I'm being unreasonable until I put myself in other people's shoes. For instance, you said you don't have a job due to depression and anxiety. This alone could be hard for a lot of people to live with (especially if they have no experience with it). Knowing that they have extended a roof over your head without them asking for money in return tells me they do care, just not the way you want them to care. It's really hard for me to give you any sort of advice on that because again, I don't live with you.

She finks on me, tells her parents all about my angry and upset Facebook ranting.

Saw that coming.

They say that they think that since they're hurting me, I should find a new place to live.

Again, they've given you a warm bed and a roof over your head. I would highly consider you tilt your personal perspective on the situation and see things from everyone's point of view. Communicate. Talk to them, not at them. People (especially adults) do not like to be talked at, or told what they can or cannot do in their own home. Be reasonable. Be calm, cool and collective. Communicate, communicate, communicate.

The entire time I've been here, I've tried to be helpful, I offered to buy my own food, do any cooking or housework they wanted.

I think this sentence is more along the lines of giving yourself an excuse for your actions, and I mean this in the most non offensive way possible.

I bit my tongue when they got crazy religious. I let them go on and on about how Obama is ruining the country. I kept quiet, did everything they asked. I ask them to do one thing for me, and two days later, they tell me I need to leave.

That's not how that works. It usually takes multiple recurring events for someone to kick another person out. And to be honest, they don't have to adhere to any of your requests. Be mature about it. Just like you, realistically, and retrospectively don't have to adhere to any of their political or moral beliefs, they can/and will do the same.

I'm debating going upstairs and taking every pill they have.

Eh, I wouldn't suggest that. Again, relax. Breathe. Think rationally for a second before you make another decision you may regret.

Also, suicide is a permanent solution for temporary problems. Temporary. Meaning the problems will pass. Suicide, not so much. Think about it before you attempt anything.

Everyone here cares for the wellbeing of our community, and I do hope that things turn out for the better for you. As for now, I would suggest openly communicating with them and figuring out a plan of action for your own life (so you can live your life independently and freely). If I lived in the area, I would offer you a place to stay, but I don't. I live in California. I'm sorry that your situation has turned ugly, but be optimistic. Keep positive, please. You can and will overcome all of this.