This thread should probably be merged with the other one, but here goes...
I know that to be here for longer than 3 months, which, I already have (am I in trouble here?),
Nope. You could be if you willingly told the authorities that you were doing this - but in all honesty, they have no way of checking when you entered/exited the Schengen zone. The exit checks aren't comprehensive enough - so you can tell them any old nonsense and they have no way of checking.
But strictly speaking, yes, you're in trouble. Although I'm sure I read somewhere that British passport holders are given six months free access rather than three for the rest of non-Schengen EU. The golden rule is that if you ever get questioned about it, then simplytell them that you're on holiday.
I need an EU short-term Polish residency.
Unless things have drastically changed, you basically get three months temporary registration. After this, you go back and obtain a five year residence permit. Or you can continue to register for three months at a time - it's entirely your choice. The system is a bit of a mess, because it seems to overlap with the non-EU system - but basically, you get automatically granted a permit to stay.
Is there anything missing, that I should be including?
This doesn't sound right at all. I don't have the form to hand from Poznań - but I simply had to fill out a small form and bring the owners of the flat along, plus providing photocopies of my passport. There was certainly no proof of income required, nor health insurance. And the Polish bank account part sounds like nonsense to me too - they certainly didn't ask me for any.
I suspect that you might have been confused for a non-EU citizen, actually.
As for the Zameldowanie - the temporary three months registration is your zameldowanie. This makes me think that you might've went to the wrong place and were trying to apply for a non-EU Karta Pobytu - which would explain all the forms needed.
I have also been told, that instead of supplying a bank statement as proof of income (because I don't yet have a Polish bank account), I can instead supply, a formal letter of employment to them, from a company I work for in Poland.
This is what convinces me that you've been trying to apply for the wrong thing. There's absolutely no need to prove anything work-wise when registering as an EU citizen - because of the EU freedom of movement, you can move to Poland for whatever reason you like. You certainly don't have to be employed to live here if you're EU.
My school can produce this letter of employment, but they assume that they must also include a copy of my contract of employment with this. This may be true so I am assuming it is.
Usually, a contract is required for non-EU in order to gain a work permit/residency. But again, it isn't required for EU.
The problem is, I don't have a NIP no. at the moment, so I am pretty sure I cannot be paying TAX, even though my school are collecting around 18% "TAX" from my wages - how?
Very common scam, it seems. The best bet is to get yourself down to the tax office sharpish and get a NIP - you don't have to be registered in order to get one, as you can be carrying out business affairs in Poland and wish to declare it without actually living here.
Once you get the NIP, ask the school to provide you with evidence that the taxes have been paid. You'll probably find that they've been creaming the 18% into their own pockets - though the usual trick in this respect is to tell them that you're taking your contract to the tax office because they want to see it. If they **** themselves and tell you not to do it, then you know they're upto no good.
When I arrived in Poland during my first few days, I was told that in order to apply for a tourist VISA for Russia, I needed to be a Polish resident, which means I had to obtain Polish Residency and live in the country for at least 90 days.
I wouldn't put it past the Russians to have such a ridiculous requirement.
Basically, I can see that you're in need of some advice here.
First step, go to the Urzad Miasta (or whatever the correct spelling is!) for your city (Warsaw, presumably!). This'll be the place where you go to register your address - and if Warsaw is similar to Poznan, then there'll be a dedicated office there for the registration of EU nationals. I'm not sure what the Polish name is, sorry - but someone will surely tell you this.
Once you've got this form, and it should be a very simple form, then you have to get your landlady to agree to register you. This is the difficult part - but I'm sure someone can advise on the best way to convince her into doing so.
Make sure that you get given a PESEL number when you register - it's not required, but makes life much easier for you.
During all this nonsense, you want to obtain a NIP number. They probably won't care less about proof of employment/etc because of you being EU - they didn't in my case, anyway. But the number should come through after a couple of weeks, which will allow your school to start taxing you correctly.
But - and this is the important question. Do you know what type of contract you have? There are others on here who can advise as to the different types of contracts - but from the sounds of things, your school is effectively pretending to tax you 18% while creaming it off for themselves. It's a common scam, unfortunately.