I am going to be perfectly blunt here enjoi, and I don't think you want to hear this, but I think you owe them this penalty money.
You signed a contract. Your contract stipulates that if you break the contract, you pay a penalty. You want to break the contract, you should be held to the penalty.
A broken hot water heater and expecting you to show up for training sessions don't really count as a breach of contract on their part. Certainly not after less than 2 weeks in country. It doesn't really sound like a real effort was made on your part to try and work with them, you just decided you didn't like it and that you were gonna bail on the job.
And yes, it is your prerogative if you just don't like it and want to leave, but you signed a contract and you need to be prepared to abide by it. Nothing will piss people in China off more than a foreigner coming in acting like the rules don't apply to him and trying to throw the weight of the embassy around. Paying 7000rmb breach of contract penalty is not "extortion" it is paying the school back for the time and expense involved in bringing you over.
Again, I am not trying to be harsh here, but I hope that future China newbies read these threads and realize that there are consequences that happen when you break contracts. The school can't hold your passport, but if they want to take you to court and sue you for the breach of contract penalty, they can. The embassy may help you to a degree, but remember, they hear stories like this all the time, and they don't have to automatically come down on your side just because you're a US citizen. The job of the embassy isn't to make it easier for foreigners to breach their contracts in China with no repercussions.
Personally? I'd find a way to pay up, or tell them you don't have the money and try to negotiate a smaller penalty. Get into a hotel in the meantime. This might be difficult without your passport, so you're going to have to cooperate with the school somewhat and tell them you will vacate the premises but you need your passport in order to check into a hotel. Do you have a copy of it at least? By making this issue antagonistic, you've created a lot of hassle for yourself.
In the meantime, do as others have suggested and contact the entry/exit bureau of the PSB and ask what you should do. Explain things from your point of view and see if there is a way out. Wait to hear back from the consulate and see what they have to say, but do not hold your breath that they will be able to offer some sort of free pass or will be able to threaten the school into complying in any sort of meaningful way.