I am in Poland under the visa-free regime. I crossed the border on 12 February. 90 days pass on 10 May. Can I stay in the country until the end of the epidemic without changing the basis of my stay?

Yes, you can. If on 14 March 2020 you were in Poland under the visa-free regime and the date of your legal residence on this basis ends during the period of an epidemic emergency or epidemic, you may legally remain in Poland until the 30th day following the date of cancellation of the last state. Your legal stay will be extended by law, and you do not have to complete any additional formalities.