Here's my answer to the questions in the OP. Firstly, the Finnish government might have well accepted the demands. It was a political decision made under duress, after all, and even slight and late changes to the OTL might sway the decision towards acquiescence. Famously even Mannerheim supported accepting the demands because he thought Finland was in military terms too weak to successfully resist the USSR.
Secondly, in the case of the actual war, I can't see the Soviets do that much better without a POD or several before at least summer 1939. After all, come autumn/winter 1939 all the major things that made the Finnish Army able to resist the Red Army would be pretty much the same, and thus the Soviet forces would initially face the same brick wall they did IOTL.
The only way for a quick victory with a late POD I can see would be a combination of the two issues: Finland accepts the demands, causing internal political turmoil and and a notable drop in national morale; and then the USSR follows up with a quick, decisive attack when the Finnish troops have just started to withdraw in early/mid November. In this case a Soviet victory before January 1940 would in no way be impossible.
I just outlined a possible timeline for Finland accepting most of the Soviet demands and then being "peacefully" joined to the Soviet bloc in the same fashion and roughly same, slow timetable as the Baltics in
another thread.
But a quick attack might have also been quite possible. IOTL, the discussions between Stalin and the Finnish delegation in the Kremlin ended on November 9th (de facto, though the Finns didn't immediately know that but waited in Moscow until the 13th for a new summons to Stalin's presence). Now, it is known that the Red Navy on the Baltic had already received orders to be ready for war against Finland on 3rd November, and that Voroshilov ordered the Leningrad military district to start war preparations against Finland between November 11th to 15th: at this point the decision to start war against Finland on November 30th had already been made.
Stalin being the cautious gambler he was, I think it is more likely that the armed forces would stand down for the while and the Soviet actions on Finland would follow the same path as with Estonia, with a slow eroding of the national will to fight by various diplomatic and military hijinks and an invasion in summer 1940.
But by most accounts the Soviets were ready to go to war whatever the Finnish answer was. Therefore if it suits Stalin's fancy, he might well order an immediate invasion even if the Finns accept the demands. In this case, the Finnish government and army would be definitely in a poorer position they were when the Winter War begun IOTL, and it might lead to a reasonably quick defeat.
So when you think about the butterflies and knock-on effects, it would be good to take in to account both of these options, the immediate, successful invasion and the "Baltic path": necessarily, they would have affect Stalin's actions and policies in 1940 in different ways.