What if Roosevelt had won the Republican nomination and had managed to beat Wilson in 1912? What would have been different under the Roosevelt administration, as opposed to the Wilson Administration? And how would it effect the 1916 election?
He would probably have got some reform measures through, but fewer than Wilson, since a Republican Congress elected on his coat tails would have been more resistant than a Democratic one elected on Wilsons.
Regarding WW1, the big question is whether his presence deters Germany from renewing USW in 1917. If so, America probably stays neutral. If not, (imho more likely) TR sends the war message in Feb 1917 rather than April. [1]
In 1916, he is probably re-elected, but presumably dies in office in 1918/19. As to the VP, your guess is as good as mine, but Harding is as likely as anyone.
[1[ Possible butterfly if TR gets entangled in a war with Mexico, so worries less about Europe