Whether or not your lawyer friend comes up with anything, file immediately for unemployment benefits. DO NOT wait another day. They'll determine if you have covered wages, if you are able to set up a claim based on the last two years, which will be your covered quarters of employment.
Then they will determine if your employer, this last one, had a valid, job related reason to terminate you. It doesn't really sound that way, does it? Sounds more like they decided to lose you for some reason, and elected to do so. You showed up, you did your best, you did not quit the job, and usually under these circumstances, they'll approve this termination. They may go back and look at the last place you worked, determine if that was a good job related reason to quit. But either way, let them do it. It costs nothing and there is no downside to working this out, seeing if you are eligible for unemployment.
Filing for unemployment does not connect with, does not help or exclude, and is not related to other labor issues like being paid correct wages, filing an EEOC complaint, whatever. It is just the recourse that you have when you are terminated by an employer. In order to keep you from receiving unemployment benefits, which were paid in by the employer's taxes, not your own, your employer would have to show that you were terminated for a genuine work related misconduct reason. They probably do not have this, from what happened in your eight days with them.
In the unemployment system, you do NOT have to argue a case or convince anyone, they will already know exactly what to do and what to ask you, and you do not wait to determine if you "might" qualify for unemployment before you file. Even if you are approved, the employer you have had last will have a very small part of your wages that will be included in the claim. That's why they love to terminate employees very quickly if they aren't going to keep them. And they probably can't keep you from being approved for unemployment, this is your ONLY recourse when terminated most of the time, especially for a quick termination like this. So file for unemployment, and move on.