re: internal Vatican politics...
I don't think the succeeding Pope would be a significant departure from the conservativism of John Paul II, since no Pope since him in OTL has been either(Francis is just a bit hipper in his personal style, but is still continuing the general thrust of Leo XIII-style economics combined with social conservativism that has held sway since forever). I suppose some people might argue that the church today is so conservative because JPII made all these right-wing appointments, but I don't know how well that holds up to scutiny(even before JP's rise, Paul VI had refused to liberalize the teaching on birth control, for example).
As for the wider political scene, if the assassination was ordered by the KGB, and if they were really thinking that it would make a difference to the Chuch's position in the Cold War, much less to the power of the West generally, they were almost certainly in the feverish sway of Great Man Theory. The Church wasn't anti-Communist because the Pope was a Pole; it was anti-Comminist because Communists are against the Church. I know JPII is thought to have played some role in the revival of Polish religious-nationalism, and that may very well be true, but it was also only one small factor in the downfall of Communism in Europe. The Kafka scholars and Frank Zappa fans who challenged Communism in Czecholslavakia likely weren't taking their orders from Rome.
And I'm not neccessarily disputing that the KGB ordered the hit, just that if they were really expecting it to be a big boost to the immune system of Communism, they were mistaken.