Its all about the setting...
Range: Legolas without doubt
Sword vs Axe: While Lagolas is fast and would dodge "easily" Gimlis axe, Gimli would withstand many hits from Legolas sword. Gimly is strong and can fight with much heavier armor and a great axe. One blow of Gimlis axe would surely wound Legolas so much that the second thrust could be the fatal one. Legolas would become so dizzy and off balanced by that powerful blow and give Gimli the upper hand. With Legolas knowing this, and Gimli beeing far from a novice, I think it would be a long fight. Mostly with Legolas avoiding Gimlis axe and trying to find openings for small strikes between the much needed parries. While Gimli on the otherhand, with his remarkable fighting skills, stubbornness and endurance never would grow tire.
I Would put my money on Gimli, think I would win the bet at least 60% of the times.
Bare knuckle fight: Gimli would wipe the floor with the slender elf
I personally favour Gimli, if he could get within close range especially so. The use of an Axe against an unarmoured Elf would be quite devastating. Plus, Gimli openly wore a coat of what I assume to be Dwarven steel mail. Whilst I doubt it would turn a direct hit with an arrow, it should certain turn a glancing shot.
Although the sprinting ability of the Dwarves seems to be, at best a concoction of the script writers for the movie. I scanned through a few of the books, and the quotes about the Dwarves and their ability seem to lend lend themselves more to the idea that they have a very high endurance and are thus, suited to long distance rather than short distance sprints.
In the Silmarillion, on the making of the Dwarves in chapter "Of Aule and Yavanna"
'Aule made the Dwarves strong to endure. Therefore they are stone hard, stubborn'
and
'they suffer toil and hunger and hurt of body more hardily than all other speaking peoples'
There's quotes in Fellowship and Two Towers which also allude the the Dwarves capacity for endurance, but I am already deviating from the point so wont bore you all by posting them. Unless they are required of course, then I would be glad to.