I wanna ask you if it's possible at the Kantors or at the banks to exchange a quite big amount of PLN to USD cash, something like 4-5k PLN in USD.
Absolutely. You can change as much cash as you want at kantors, although it would be wise to make special arrangements for any serious amounts of cash. You can even negotiate the rate online with some kantors - I did this just yesterday, and got 4.13 (when buying Euro) when the interbank rate was 4.12005. If it had been a serious amount of cash, they were willing to go down further.
Banks will not really negotiate on the rate unless you're transferring regular large amounts of cash (i.e., a minimum of 10k EUR on a regular basis).