WI Teddy had been killed?
Fascinating possibility. At the time, John Hay was secretary of state and would have succeeded TR by the laws of succession. Hay was rather advanced in years, though, and I doubt he would have wanted to be any more than a caretaker for the balance of the term.
That would have likely paved the way for Elihu Root as the GOP nominee for 1904: he was progressive enough to appeal to that wing of the party to a degree, yet his ties to Wall Street would have made him attractive to the conservatives. Paired with someone along the lines of Missouri governor Herbert Hadley, or Indiana senator Albert Beveridge, it's reasonable that Root could have won the presidency.
With Root in the White House, it's possible that Taft could have been persuaded to come home from the Phillippines to take a seat in the cabinet, and later on the Supreme Court, thus avoiding the presidency altogether. I could see either progressive VP Beveridge or Root's secretary of State Charles Evans Hughes as the 1912 GOP standard-bearer; while neither was as dynamic as TR, I suspect that a united GOP in 1912 could have beaten Wilson.
When one gets to the war in Europe...I'd guess that there would have been a statement of neutrality by President Hughes in 1914, but the sinking of the
Lusitania might well have drawn in the United States, the issue of armaments on board covertly notwithstanding.