How outdated is the [1D] Mark III compared to other Canon or Nikon cameras in the low/mid price range?
A bit. But in other ways, not at all. The biggest advances since 2007 have been in high ISO performance and resolution on the sensor. Comparing a 10MP 2007 APS-H sensor to a current 20MP ASP-C or full frame sensor is going to be a bit like comparing apples to oranges. But generally, if you've wanted to print larger, shoot above iso 1600 with acceptable levels of noise, or were looking at newer features like video, wi-fi, or off-camera flash with the new RT system, then maybe a new body would be worth it.
What newer canon model can be compared to the Mark III? The high-end cameras will obviously beat the 1D Mark III.
1DX or the 7DMkII are liable to be the closest current "successors" to the 1DMkIII. But the 7DMkII, obviously, will lack the build quality, the integrated vertical grip, weathersealing, and some of the physical UI stuff that makes the 1 series the pro line. The 1DX is full-frame--Canon doesn't do the APS-H 1.3x crop thing any more. And the 7DMkII basically has a 1D AF system in it these days. So if fast-action is why you got a 1D, then maybe you could step down to a 7DMkII.
Would you even replace the Mark III having all this equipment?
As I stopped using my Canon kit (50D/5DMkII) most of the time in favor of lighter and smaller mirrorless cameras, I wouldn't even have bought a 1D in the first place, so I'm probably the last person to ask. :) But generally, I think it depends on what you use your 1DMkIII for, and how the features that are specific to that camera are used by you, and whether or not you're dissatisfied or frustrated with the camera body. Whatever those frustrations are should lead you to what it is you want in your next camera body. But if it's working and you're happy with it, and you see no reason to swap, then there's no reason to go there.
If yes, what would be a low/mid range alternative?
Personally, I think a 7DMkII or 5DMkIII/6D (although waiting for a 5DMkIV to catch up Af-wise with the 7DMkII is possibly wise if you shoot a lot of fast-action stuff; the 5DmkIII is nearing the three-year mark, and most of the 5D models have been refreshed at that point). And the 6D would be assuming you're not shooting a ton of fast-action at all, as its AF system is pretty primitive. I wouldn't recommend going from a 1-series to a dRebel (XXXD or XXXXD), simply because you're probably very very used to dual-wheel and a top LCD and joystick. Maybe a 70D if you can give up some of the AF system and the joystick. But given that all your lenses are full-frame, I'd actually say that a 5DMkIII is probably the way to go--if you go with a 1.6x crop body, it'll feel a bit like tunnel vision, and you might want an EF-S lens (10-18 or 10-22) for the ultra-wide end.