Does this verify Molotov's claim that the Soviet union only wanted to secure the area around Leningrad?

Well, just USSR asked "just a tiny border changes and certain military bases" from Baltic states as well and we know what happened to them.

Plus Molotov-Ribbentrop pact clearly shows Nazi Germany and USSR agreed that Finland was part of Soviet "sphere of influence" and neither side wouldn't interfere with other side's actions towards those nations. Every other nation in that sphere of influence was occupied by USSR.

-> Finland was in same position as Baltics and was first tried to lure into same trap as Baltics, only difference was the fact Finland chose the military way to try to secure it's borders.

When the Soviets finally broke the Mannerheim-line why did they not continue to get all of Finland under their control? The Finns should have been pretty much broken at that point?

In early 1940, Red Army had fairly poor idea of the Finnish reserves and strengths. Hell, even the Mannerheim line was partly a surprise to Red Army high command, though building it wasn't even a secret operation. Red Army espionage at the time was fairly terrible, and the bureaucracy made sure the information moved slowly to the top. Finns indeed were broken at that point, but I doubt Stalin had a full picture about how broken.

Western Allies were planning to send tens of thousands of troops to aid Finland, meaning it's pretty serious intervention. Stalin's greatest fear in the situation must have been Western Allies uniting with Nazi Germany against USSR, which was to be avoided with all costs... The war with Nazi Germany and USSR was clearly inevitable, two ideological enemies.