14th century craftsmen could not make a floating axle treb on that scale I don't think. Making the wheel assemblies basically requires rail road level steel working. If you want to go in that direction, I recommend what I saw one college team did with their floating axle, it slid on the tracks. You could include rollers to reduce friction. However, given large scale light wood, they could make a treb basically identical to Trebarbaric where the frame rolls across the ground quite a lot when throwing (That thing is so amazing to see in person!), except 10 times bigger. Having the frame move instead of the axle is not only much easier and more realistic, but you gotta admit having a frame that size roll around is freaking cool. The frame moving has basically the same benefits of the axle doing so. Check out my videos of it.
I should note that there was a steel FAT that threw pianos 900 feet with 9 tons of counterweight. They can be made large, but it is much harder to make a large one technologically than a wheeled HCW like trebarbaric. Also, in a FAT, the arm is not the big deal part as it is with most other designs. The FAT's track, vertical stand, and counterweight holder/axle are the issues, not to mention the much higher complexity.
Trebs scale basically linearly, and mass scales independently from size. So 10 times as big means 10 times as far of range. Now, because of air resistance, it would not throw 18,000 feet (thats like 3.4 miles). The main issues preventing such scaling is the availability of a throwing arm for such a machine (to be strong enough, it would be too heavy), and the budget man power and space. Seeing how you have actually addressed every one of these issues, you should be able to keep things quite realistic.
On a technical note, you want a stiff, not flexible arm (The arm flexing just removes energy from the throw).
The specific parameters really depend on its intended use. Trebs were exclusively anti castle weapons designed to slightly out range the archers. Thus they threw 100-300 or so pound stone balls 200-300 yards or so. (Those numbers are just guesses, I haven't studied hurling history really) Any more weight or range was useless.
The pay load could be anything you want. As I said, mass scales independently from size. Also, you can have a counterweight to projectile ratio from about 10:1 to very large (Trebarbaric has 8-10 tons for an 8-10 pound projectile, thats 2000:1). Assuming access to large amounts of Icewood and lead are available, such a machine could, to be true scale with medieval ones, could throw 10 times as far with 1000 times the counterweight and projectile mass. That means 100 tons going 5000 feet, and maybe a 8,000 ton counterweight. That is kinda insane, why would anyone need to do that? It would throw about the same distance with lower masses, say 5 ton projectile and 400 ton counterweight.
One on your scale could be used for shipping (lol), anti personal (lots of lethal sized rolling projectiles), building smashing, corpse chucking (get them with the black plague! That was actually used at least once), fire spreading, or basic scaring the heck out of every one. If you can offer up their reasons for building the device, I should be able to work out some pretty reasonable specs for it. It makes a huge difference what they want to throw. If you just want to throw something huge (say, a 10 ton block) 300 yards, you really have no need to make an arm that long, you just need lots of counterweight. I would say we are looking at a range of about 3000 feet minimum here (assuming dense projectiles), but more or less could be possible depending on what projectiles were chosen based on the intended use.
If you are willing to mess up and kill some people (oops), a machine that size could throw people (the accelerations on large machines are not higher, they just accelerate over a longer time as the counterweight acceleration is fixed to gravity). Being large, it could throw people at 400+ mph a few thousand feet. Paratroopers! Actually a machine that size, if tuned to throw small dense things (100 pound lead balls) and loaded with 50-100 tons of counterweight or so, it would probably have issues due to throwing super sonic.
We are going to have to assume Icewood can be pounded out and woven into ropes to make the sling, because they did not have modern synthetic rope, or high strength steel cable. You have to make the sling and ropes to pull the arm down for cocking out of something.
For construction and proportions, copy trebarbaric. It is a simple and good design. For lifting parts into place, make giant A frames, which get up righted and used like cranes (There is a great nova on trebuchet construction using traditional methods that covers this - some one stuck part of it on youtube, http://www.youtube.com/watch?v=j9lhcjupnzk I think you part part later in the show though)
For reload time, I honestly have no idea. Assuming an easy to setup winching system, and unlimited man power, the thing could be cocked pretty quick (maybe 5 minutes after they get some practice). The loading time depends on the projectile, but would probably take quite a while. Also, getting the winching system setup would require a brave guy to actually climb a good 500+ feet up the thing (probably still swinging a bit!) to attach the draw down lines. Waiting for the machine to stop swinging before reloading would take a long time, and any effort to seriously reduce the time by adding friction would just catch the thing on fire. I would say at least a a minute. Pushing the frame (assuming its a rolling HCW) around would take a while too. Overall, I don't think throwing more that once every 15-30 minutes would be practical, but this is fiction, and 10 min would not be pushing it too much.
So, what what is is going to be used for? Then I can whip up some numbers for you.