Should I believe him?
No. They are vastly underestimating the cost of the apartment, and not telling you about the hidden costs. First of all, good studio apartments (one-room apartments) are very hard to find. They are very much in demand, and the supply is not so good. Most of them that I have seen have kitchens that are basically useless, so that means that you will have a hard time cooking at home. A good studio apartment is going to cost from 1500 to 1800 PLN all inclusive (rent, hidden administration fees, and all utilities except internet/phone/TV). If it costs less than that, there is something wrong: bad kitchen, bad windows, bad location, bad neighbors, etc. That is why you need the help of an OLDER experienced Polish colleague to find a good apartment.
My guess is that you will not be able to find a good one-room apartment and will have to rent a good two-room apartment, which are much easier to find. The total cost for that will be between 1800 and 2200 PLN, all inclusive.
If you are making 7500 PLN gross, that means about 5000 PLN after taxes. After deducting the cost of your relocation, your apartment costs and your living expenses, the most you would be able to save is 1500 PLN, and that is if you live like a monk. You won't be satisfied with that lifestyle for long and so you will dip into those savings to avoid going crazy. Even then, it will not be a comfortable life.
Like I said, if you want to save 3500 PLN a month, you will need to earn 12,000 gross. I doubt an employer that initially offered you 3000 PLN is going to go that high. I would never work for an employer who made an initial offer of 3000 PLN. They are obviously either unethical, or just plain stupid, and it is not pleasant having either as an employer.
Again, if you are interested in saving money, Poland is not a country you should even be considering. Wages are very low, cost of living relative to wages is very high, and savings potential is downright abysmal.
You will never find a good job if you depend on recruiters and internet sites. Good jobs are advertised exclusively face-to-face in the real world. Recruiters and internet sites get the table scraps. If you want a good job, use your network of real-life contacts to find one, especially those people who have worked in richer countries where the savings potential is much higher than in Poland.
This job is not the dream job you are looking for. Stop wasting time on it and look for better opportunities elsewhere. They certainly exist.
As for pollen in Poland, it is a big problem. Often, there is so much pollen in the air that it forms patches on the roads and covers cars with a thick layer of "dust" after a rain. I was surprised when I first saw it. If pollen is a problem for you, then anywhere with wet weather is pretty much out of the question because of the abundant vegetation. Try to find a job in Saudi Arabia or the Emirates.