I think an invasion of Korea was very likely regardless of whether the Oda or Toyotomi unify Japan. There were an absolute crapton of restless, veteran soldiers (both samurai and commoners) running around after the hundred years of endemic warfare that was the Sengoku Period, and it makes sense to give them an external enemy to fight instead of letting them fester at home and potentially think about turning on their new overlord because he doesn't give them the exact fief they want or whatever. I'd imagine Korea is a more attractive target for conquest & settlement than cold, remote Hokkaido as well. That said, I'm not sure if Nobunaga would be as insanely ambitious as Hideyoshi, who infamously wanted to use Korea as a platform from which to conquer China & India - Hideyoshi may also have been motivated to aim so high by a need to further prove himself due to his peasant background, which obviously wouldn't have been a problem for the proudly Taira-blooded Nobunaga.
I would agree though that Nobunaga coming out on top overall gives Christianity a big boost in Japan. As has been said, he was fiercely hostile toward Buddhism but very welcoming of the 'Nanban' and their ways, even if it was probably more-so because he appreciated the firepower they could sell to him rather than any genuine spiritual interest in the Gospel (a Portuguese missionary who met him, Luis Frois,
pretty much described him as an atheist). At the time of Nobunaga's death he wasn't yet in control of far western & eastern Japan: in the former area the Mori clan of western Honshu and Shimazu of southern Kyushu were still holding out, but both conveniently had an enemy in the Otomo clan of northern Kyushu, who were among the highest-profile Catholic converts in Japan and historically aided Hideyoshi in subduing their rivals.
If Nobunaga, famous for his much more brutish and bloody-minded temperament compared to the compromising Hideyoshi, ends up destroying those rival clans as he did to many others that crossed him previously and heavily favoring the pro-European Otomo as his agents in far western Japan, then coupled with his tolerance of Christianity I could imagine Kyushu at least becoming a Catholic-majority bastion in the medium term. It's more difficult to imagine Christianity becoming
the majority religion in all of Japan, especially in the much more traditional-minded center and east of the country, but I think a case could be made for it eventually dominating the western provinces and becoming as much of an influence on Japanese culture overall as Buddhism.
Also, if Nobunaga lives at least as long as Hideyoshi, he will have another heir of age -
his grandson Hidenobu, son of his oldest son who dies at Honno-ji, who will be 18 in 1598 and was historically the Toyotomi's favored candidate to head the Oda clan. At least none of his potential heirs will be underage, unlike Hideyoshi's. But yes, it does seem quite plausible or even likely that the Oda Shogunate would tear itself apart in a dynastic civil war between Hidenobu and his uncles Nobukatsu & Nobutaka once their founding patriarch croaks (this time from natural causes). Assuming the Oda don't fatally weaken themselves in this episode of bloodletting and get overthrown by some other daimyo, I do think that Japan would remain open, tolerant of Christians and inclined toward foreign trade & expansion. Maybe an Oda Shogun will even make a real go at conquering China if & when the Ming should collapse as they did historically.