The publication fee will be covered only if a Freie Universität member is responsible for the settlement of the fee as "submitting" or "corresponding author." The author must be a Freie Universität member at the time when payment is made (§ 43 Berliner Hochschulgesetz).
Yes. Everybody officially pursuing doctoral studies at a university in Berlin is a member of that university according to section 43(5) of the law governing higher education in Berlin (Gesetz über die Hochschulen im Land Berlin, BerlHG). Consequently, you can apply, as long as your article is in keeping with the conditions set forth for funding by the German Resarch Foundation (DFG).