1) Do you need to have written agreements with affiliate and advertising companies (US based) in order to receive wire transfers from them and justify your income? Or printed out terms and conditions will work?
I'd be surprised if Poland cared - as long as you can show how you gained your income, there shouldn't be any issues. Certainly, I'd be hugely suprised if you needed anything to accept wire transfers - just that if you earn a huge amount from a source, they might ask you to explain where the money is coming from, but that's the same from any country for any amount of money.
2) Do you pay VAT on such kind of income?
I don't think so, but only the Polish taxman will be able to tell you for sure. But I can't see why VAT would be charged unless you're actively selling space on your websites.
3) How often can polish LLC pay out dividends: yearly, quarterly, monthly?
No idea, I'm afraid.