So if Roosevelt was President in 1914, and was able to negotiate with the various nations involved, and World War One doesn't happen, what happens from there? Does Europe simply end up with another, far nastier war in the 1920's or 1930's? And what happens with US politics during/after Roosevelt's third term? I'm assuming he doesn't get a fourth.
Good question. It all depends of course on the terms of the solution.
If Austria sees every other Power aligned against them and Hungry opposed to going to war as it was IOTL, it has no choice but to back down. What happens then is the key. If the humiliation is great enough then it may even lead to the collapse of the monarchy. If that happens, then Willies power becomes shaky in Berlin since his power will be seen to be less than inevitable and more power will gravitate to the parliament.
Unless there is a permanent solution in the form of machinery to resolve the underlying and chronic problems in Europe a future war appears inevitable. Just how much the governments in Paris and Berlin in particular, but also in London and Moscow, are willing to allow arbitration to be binding is the point. The agreements reached on arms limitations and other issues just prior to the outbreak of war shows these governments were willing to go down this path.
I think TR and his influence is somewhat overrated. Why would Austria and Germany just agree to accept his arbitration? The US has no military, diplomatic or economic power to back up any demands. His negotiated peace on the Russo-Japanese war came when Russia was more than willing to accept virtually any terms that would get it out of a sticky situation and Japan was in no position to carry on the conflict and had achieved its aims anyway. Both AH and Germany were supremely confident that they would win any war they embarked on.