Is it possible to move to another country (further West) without a visa?

As a general rule, if you would like to enter another country, you should contact its embassy for a visa. The fact that the Commander of the Polish Border Guard allowed you to stay in Poland for 15 days does not give you the right to travel to other EU countries. If, however, you entered Poland on the basis of a visa / visa-free travel rules (for Ukraine), you may travel to other EU countries for 90 days.

If you apply for international protection in another EU country, then, according to EU law, you may be sent back to Poland where your application for international protection is to be reviewed. This does not apply to people who benefit from a visa-free movement, their cases should as a rule be recognized in the country where they submitted an application. There are also other exceptions to that rule, e.g. if you have close relatives in that country or if there are humanitarian considerations. As of now, we do not know what practices will EU countries apply.