The Stalin Note from an East German standpoint was extremely important. Stalin made the overture of a reunified Germany, which basically turned the Socialist Unity Party (SED) leadership of the GDR on its head. Only four years earlier individuals like Walter Ulbricht and Wilhelm Pieck, who would become the first and second leaders of the GDR, were personally sent to Germany to formulate a new communist government. The Stalin Note went against everything he had promised and done for the SED.

The German reunification under communist control was a long-shot, likely impossible, which both Stalin and the SED knew. USSR support was the only thing keeping the GDR's government in place, as they'd later learn during a later uprising that had to be quelled by Russian tanks. The sentiments of reunification mentioned in the Stalin note likely are what empowered Lavrentiy Beria, Stalin's head of the secret police and candidate for succession, to take up a similar position. Likely, he felt it was a safe plan and that nobody would turn against Stalin's will. He made a few anti-GDR statements, even going so far as to indicate that he didn't even care if the FRG took over the eastern half of the country.

When the Malenkov/Khrushchev/Beria power struggle broke out in the Communist Party of the Soviet Union (CPSU) following Stalin's death. Beria's comments on Germany were bolstered by the ideas of Stalin and the SED leadership were deeply concerned by his sentiments, much less that he might potentially take over the USSR. They were essentially relegated to sitting back while this power struggle played out in Moscow, with the very fate of their support from Russia hanging in the balance. In simple terms, without the Kremlin's continued support, there would cease to be an East Germany. Economically, politically, militarily, etc. they were almost entirely reliant on the USSR.

Fortunately for Walter Ulbricht (SED leader at the time) and his loyalists, Beria lost the struggle. His comments on Germany were the main impetus for the CPSU Politburo turning against him, with Khrushchev, Molotov, and Malenkov leveling the accusations that Beria was attempting to sell out the GDR to the West. Beria struggle for leadership would come to a swift end when he was formally condemned, denounced, and arrested in a dramatic episode in the council chambers when living legend Gen. Zhukov would personally detain the man.

With all that out of the way, it would appear that Soviet leadership was serious about the note, enough that Beria (a shrewd man) backed that horse in his rhetoric for years. The rationale is difficult to determine, as it was a proposal straight from Stalin. Likely the USSR was still reeling from the war and it was the safer path, rather than directly confronting the West. After all, they had been allies of necessity rather than choice. However, being Stalin, it very well could've been a ploy to fool the West into seeing him as a more diplomatic individual who wanted to cooperate. Perhaps he'd make a proposal he knew they'd reject, but one that made it look like he was cooperative.

I can't speak much for the West's reaction, as that isn't my field. Sorry, there's a lot in your question, so hopefully that gave some good anecdotal information that can contribute to your overall inquiry. At least in my opinion, the fear of the SED and the rhetoric of Beria prove that it was serious and came with mixed emotions from all sides.

See: Ulbricht: A political biography (Carola Stern), the official transcripts of the CPSU Beria denouncement, Anatomy of a Dictatorship (Mary Fulbrook)