Does a Ukrainian student who was in Poland before 24.02 on the temporary stay which “ended” but was automatically extended by COVID and now by the war, can travel to other EU countries and then back to Poland?

There is no such possibility. The position of the Border Guard authority in this regard is quite clear. Such a person is entitled to stay in Poland and not in other Schengen countries. This right arises from the Covid Act as well as special arrangements under the Act on Assistance to Citizens of Ukraine in connection with the Armed Conflict on the Territory of the State. However, these solutions are national in nature. Crossing the border in such a situation may result in being exposed to the charge of violating Article 302(1)(10) of the Act on Foreigners, which will result in issuing a decision on the obligation to return.