How would this affect 1930s Japan and later, the course of the Second World War?
Several recent threads led me to begin reading Japanese history again. I don't see any changes, major or otherwise, at all flowing from Terauchi's death. The Count's role in politics, plus the conduct of the China war and later Pacific war, simply isn't substantial enough.
With regards to the war in China, Terauchi wasn't part of the Kwantung army during the '20s and 30s and it was that organization's deliberate course of provocation which forced war upon China over Tokyo's phony protest. More tellingly, the Count didn't any steps to rein in the Kwnatung when he was war minister.
With regards to the WW2, while he was the overall commander of Japan's forces fighting in southeast Asia and the Pacific, he wasn't the commander on the spot and his ability to influence the many fronts "his" troops fought on was severely constrained by Japan's tiny sealift capacity.
The factional disputes mentioned are also a red herring. Terauchi a member of the Kodoha faction, but so were many other officers in the IJA. As for the Toseiha faction, it was just as militaristic as the Kodoha. It's opposition to the Kodoha had everything to do with issues of personal ambition and little to do with national policy. Factionalism was a way of life in the Japanese armed forces and every general officer, army or navy, belonged to one or more.
Terauchi is just one pebble of millions inside the avalanche, so his death isn't going to make any real difference.