I'm not exactly sure what the main problem is here. Here's what I'm reading, tell me if I've missed anything:
1) They like your work, so they give you more hours. Presumably they pay you for those hours (you haven't said otherwise). This seems like a plus (unless you are saying you don't want those hours, in which case this seems like an HR matter; you should ask them to give you fewer hours, I'm sure your coworkers wouldn't mind the extra cash that you're leaving on the table). In the case you don't want to work those hours, you don't have to be aggressive about it. Just go to the person responsible for scheduling and say something like:
Hey, Bob. I saw I've been scheduled to work a lot of hours this week. This doesn't leave me enough time to do other things in my personal life. Can I ask to be assigned fewer hours?
See what they say. As long as you're not asking to do less work than your coworkers, but simply being asked to be treated equally, that seems like a reasonable request to me.
2) You don't mention your employer not paying you; in fact, you even mention them giving you an advance on your paycheque! This seems like a plus; even though they are paying you under the table, they are paying you minimum wage, and that (seems to be) what you signed up for. So no problem with payment.
Those are the 2 most important things: You are getting an amount of hours you want to work, and you are getting paid for the hours you are working. Now, onto the rest:
3) Your company is doing things which are (probably) illegal. This is not your problem unless you make it your problem. If they are paying you, you shouldn't care if it's above the table or under the table, by cash or by cheque or by direct deposit. You are working, and you are being paid for your work. Your concern stops there. If your company is paying you under the table to evade taxes or something, that's their problem not yours. You can whistleblow on them if you want to, but as long as their (potentially) illegal tactics aren't negatively affecting your employment, imo it's not your business or your concern.
4) This particular passage is troubling to me:
Yesterday I received an email in which the boss accused me of falsely telling him that I have received my permanent contract, "hopefully not on purpose". It is true that I hadn't received my contract, I was completely mistaken.
So you said something to your boss, he raised it as a concern, turns out you were mistaken and he was right. This doesn't seem like his problem, it seems like yours. You falsely told him something that wasn't true, and he called you out on it. Your job is to be honest with your boss. You lied to him (intentionally or unintentionally, doesn't really matter). He got upset with you for lying to him. Seems reasonable to me. Not sure what the problem is here.
5) Your boss made a comment about you being a spoiled brat. Without knowing you, it's hard to make a determination. I'm not going to sit here on the internet without knowing you or your boss and categorically say that you or he is 100% in the wrong on this one. Without him reading this it's hard to give him feedback directly, but since you are reading this and you are here I will give you feedback: have you considered the things you are asking of your employer, how much you are asking, and when you are asking them? As an example, most employers, when asked for an advance on a paycheque, would simply give a hard "no". Yours allowed it. That's already a big plus for your employer, that they went out of the way to help you out. Are you sure you're not asking too much of them?
6) In the specific example you gave where he said if an employee asks for an advance payment and then cheats him, he will ruin their career, he may have said it in a bad way, but the essence of what he said seems reasonable. If you ask for an advance on your paycheque, and then you do not work the hours required to make up that advance, you have essentially stolen money from the company. If you go to another company later and they background check you, and as part of that background check they contact this previous employer, that employer can say whatever they want about you (modulo libel, i.e. straight lying). As part of this, they may say that this employee (i.e. you) stole from the company by way of taking a cash advance and then quitting shortly after. This seems not unreasonable to me. If you take a cash advance on your paycheque, you better be prepared to work it off. He could have said it better, and he didn't need to threaten you, but the essence of what he said is completely reasonable to me.