To be blunt - you're in a bit of bother here. I'm tired, but I'll try to explain.
I was under the understanding (from the information given to me by the people who work at the foreigners department) that once I have a residence permit I can work at any school.
No. You can only work for schools that have acquired a work permit for you - unless they are approved by the Ministry of Education. This is treated very, very seriously - if you're caught, the usual punishment is deportation. It's important to separate the residence permit from the work permit - one allows you to stay, one allows you to work (for that particular company).
I went in to check the status of my application they told me I need to explain a few contracts with a couple of schools (the ones which aren't certified places of learning) and why the border police were not able to find any neighbors who knew or recognized me at my melded address.
I would strongly advise you to take an English speaking lawyer - there is a very good chance that they've already figured out that you were working illegally. The border police not being able to find you isn't so serious - but the work permit issue definitely is. Having said this - they could use the border police findings as a reason to deny your new residence permit, without getting into the technicalities of you working illegally.
The fact that they want to interview you under oath suggests that this is quite a serious affair and much more serious than the normal interviews there.
1) I have been working for 3 non-certified companies for a year without a work permit (because of the information provided) and it can be proven on my account statement which I had to give them
2) I don't live at the melded address because the place I live at is a private flat and the owners have rented it out on the DL.
1) Uh-oh is all I can say. You really, really must get a lawyer - the fact that they already know about you working illegally means that you're going to be, quite frankly, in the ****.
2) On the DL? I don't understand. In regards to not living there - mostly, they don't care in Poland - but as I said above, it might be used against you to make the case for deportation/refusal of the residence permit.
Sorry that it's not the news you want to hear - but I think that you really are in a tough situation here.
If you want, you can e-mail me at email@example.com and we can talk about it privately :)