Probably not, given the circumstances of his death and the later political situation.

After his death and subsequent semi-cremation (with gasoline, in a bomb crater, with Red Army troops closing in), the Soviets pretty quickly captured the area. Despite the fact that they actually possessed what are very likely to be his remains (they were buried and later destroyed by the KGB), it was the official position of the USSR that Hitler was hiding in the West after the war, so they wouldn't have issued any sort of death certificate.

They took this position for ideological reasons: Fascism was a logical extension of liberal democracy according to Marxist thought (this is a vast oversimplification), so ideologically the Western allies were ideologically aligned with the Nazis and were thus protecting them after the war. This shows up elsewhere in postwar Soviet propaganda as well.

Yes.

Here is the relevant answer by /u/Astrogator.