First question... do you want to shoot handheld much of the time, or are you prepared to use a tripod or at least a monopod most of the time.
If you want to shoot handheld, and want to stick with a prime lens rather than a zoom, then the 300/4 IS and the 400/5.6 (non-IS) are your two main choices. Maybe the 400/4 DO. There really aren't many other options.
300/2.8 IS II or 500/4 II are a bit smaller and lighter than the Mark I versions of each that I'm using, but still are largely tripod-only lenses. Sure, they both can be handheld briefly, but you won't want to do that for very long. Especially the 500mm. Figure a sturdy tripod... and if you want to shoot sports or nature, probably a gimbal mount. Now, both of those lenses are light enough that you can use a gimbal adapter such as the Wimbeley Sidekick with a standard, heavy duty ballhead. That might be preferable to a fully dedicated gimbal head on a tripod (which means switching heads if you ever want to use the tripod for anything else... and is pretty much necessary for 400/2.8 and 600/4).
I'd try to stick with 1.4X teleconverter, at least to start with. A 2X costs more image quality than a 1.4X. And a 1.4X costs one stop of light, while a 2X loses two stops of light. You'll be able to autofocus - center point only - at effective f8 with your 1D series camera, but not with your 5DIII (you can "fool the camera" into trying to focus at f8 or smaller, but should expect it to be slower and hunt more). An f4 lens with a 1.4X TC is an effective f5.6 and an f5.6 lens with the same TC is an effective f8. An f4 lens with a 2X TC is an effective f8.
The longer your telephoto and the more distant subject you are trying to capture an image of, the more atmosphere you are shooting through and that can cost some image quality too. Plus, even with IS, the longer the lens, the more you are dealing with keeping things stable.
It's like a tug of war.... you want the most reach, longest lens you can get... especially shooting birds. But the closer you can capture the shot, the better the quality will be.
So, in general, try to use the least strong TC you can.
You also might consider a 1.6X camera such as the 7D. That will put "more pixels on target" with long lenses, than your 5DIII.
It's sort of unfair to compare the 300/4 with the 300/2.8. The latter is one of Canon's very best lenses... might be one of the best lenses from any manufacturer. The 300/4 is merely "very good".
The 300/2.8 is designed for wide open use and sharp enough to make your eyes bleed at any f-stop. It's also got wonderful bokeh. The 300/4 is sharp, but not as sharp. It benefits from stopping down a little. And, it doesn't render as smooth bokeh (except when used wide open where the aperture is truly round). Still, the 300/4 is a super lens.
What you might consider, get the 300/4 IS and shoot with it for a while. Try it with 1.4X, which is a good solid combo. I'd recommend the Canon 1.4X Mark II or Mark III TC, or the Kenko 1.4X Pro 300 DG or DGX. Those are all quite good. I don't think the Kenko "reports" to the camera (so will still try to focus at effective f8 and f11 on most/all Canon), but since the combo is f5.6 effective, that isn't a factor in this case.
You can always add a 2X later, but probalby will find you use it less than the 1.4X, on fewer lenses.
If you find you want f2.8 lens, it's wouldn't be a big deal to sell the 300/4 for most of what you paid for it, to upgrade. Think of the purchase of the 300/4 as a long term, low cost rental! You might end up getting the f2.8 too (I did). And you might end up with 500/4 as well... not instead.
Other thought....
The 400/4 DO is an interesting lens... more handholdable. It's not so great with TCs, though, from what I hear.
The 400/2.8 IS II might be a single lens solution, instead of getting 300mm and 500mm (which I would predict you might want). Sure, it's more expensive than the 500mm... and bigger/heavier... But if you might end up with a single lens and that might be the more cost effective and practical solution in the long run.
There are other possibilities...
The Sigma 120-300/2.8 OS that Ken mentions seems a fine lens... perhaps the highest quality zoom option until Canon gets around to actually offering their 200-400/4L Externder 1.4X. The Sigma zoom is going to be a tripod lens most of the time, I'm sure. It takes teleconverters pretty well, too, from what I've seen.
It's going to cost a small fortune, I'm sure... Canon hasn't told us how much yet,. or even when it will be available... but they have given us a preliminary peek at the upcoming 200-400/4L zoom.... It will have a built in 1.4X teleconverter, so can serve as a 280-560/5.6 as well.
I use 500/4 IS, 300/2.8 IS and 300/4 IS... the 500mm and f2.8 lenses on a tripod or monopod mostly, the f4 lens handheld much of the time. It's hard to show the differences at Internet resolutions...

Sadie, after the dance.
EF 300mm f2.8 IS lens at f2.8. EOS 30D at ISO 640, 1/500 shutter speed. Gitzo 1325 tripod, Kirk BH-1 ballhead, Gitzo 1321 leveller, Wimberley Sidekick gimbal mount. Avail. light (no flash). M.Pineo & Adonis
EF 300mm f4 IS lens at f4.5. EOS 7D at ISO 1600, 1/320 shutter speed. Handheld, avail. light. Lowrider (2011 CSHA Region 5 Gymkhana, Big T)
EF 300mm f2.8 IS lens at f3.5. EOS 7D at ISO 1600, 1/2000 shutter speed. Gitzo tripod w/Sidekick, avail. light. Last barrel (2011 CSHA Region 3 Gymkhana, Cloverleaf)EF 300mm f4 IS lens at f5.6. EOS 7D at ISO 800, 1/800 shutter speed. Handheld, avail. light. Western Grebe
EF 500mm f4 IS lens with EF 1.4X II teleconverter, effective f5.6 aperture. EOS 30D at ISO 200, 1/1000 shutter speed. Gitzo tripod w/Sidekick, available light. Waxwing... all-you-can-eat luncheon
EF 300mm f4 IS lens at f5.6. EOS 7D at ISO 3200, 1/640 shutter speed. Handheld, avail. light. When you've got an itch... Blacktail mule deer (m)
EF 300mm f4 IS lens with EF 1.4X II teleconverter, effective f8 aperture. EOS 5D Mark II at ISO 1600, 1/250 shutter speed. Monopod, avail. light. Determination Peppertree Ranch Hunter-Jumper Show 2011EF 300mm f4 IS lens at f5.6. EOS 7D at ISO 400, 1/2500 shutter speed. Handheld, avail. light. The B&W image and the Waxwing above both required some work on the backgrounds in Photoshop, to smooth out the bokeh a bit. It's just the nature of the 300mm f4 IS lens that the bokeh isn't as nice and smooth as it is with the 300mm f2.8. The f4 lens has an 8-bladed aperture and the f2.8 lens has specially curved, 9-bladed aperture. The f2.8 lens is sharper wide open (but watch out for shallow DOF), while I prefer to use the f4 lens stopped down a little for subject sharpness, which means the aperture comes into play (wide open the aperture on any lens is perfectly round and "at it's best" for bokeh).