How should I follow up a second time?
Just call them up and ask politely what the status of the hiring process is. I would personally try not to call more than once every two or three days, to avoid sounding desperate.
Should I tell them about the rent, other opportunities and such?
No, absolutely not. They understand that you need money for things like rent and food, that's why we all have jobs, after all. And telling them about other opportunities after you've (verbally?) accepted their offer is just insulting.
I am concerned because I was terminated from my last job, and I am afraid this would have impacted this hiring process.
In my experience this is the sort of thing that usually comes up in the job interview ("Why did you leave your last job?") or, if not, then in the background checks that are generally done either before, or shortly after extending a formal offer.
I'm not saying it's impossible that this is the reason for the sudden silence, but it seems unlikely to me. I would expect, if they have actually decided to call off the job offer over this, that they would have said something more conclusive. ie: instead of giving a delay tactic like "we're waiting for sign off on the documents" I would expect them to say something like "we've decided to go in a different direction." If they have indeed decided to drop you, then stringing you along - especially after giving you a start date - is only postponing the issue of telling you.
the job posting was refreshed again (with lower experience).
I can understand why this looks bad to you, but there are other explanations. Perhaps they're planning to hire more people in positions subordinate to yours.
I am also concerned because I am supposed to start next Monday, and I am feeling ignored.
Given that you were fired from your last job, I infer that you are now unemployed, and therefore probably free next Monday. Assuming you haven't heard anything new by then, I would suggest going in to the office just as you have been asked as it is apparently your "first day." Perhaps they'll just have you sign the documents then.
If they don't send you the contract beforehand, they should at least have it ready on day one in the office. Be careful about starting to work without having signed an actual contract - you don't want to be doing work without legally enforced compensation! - but in my experience the first day is usually pretty laid back anyway.
Also, don't feel uncomfortable about taking further interviews if all you've got is a verbal offer. If everything works out, you can let other companies know you've accepted an offer after you've signed the paperwork on this one. In the meantime, you can let them know you've got other offers on deck that you expect to hear back from soon, and let them react accordingly.
