So this is about DaVinci Resolve since it has the most flexibility for setup, many other systems are borderline turnkey.

The recommended setup is a super micro chassis with dual xeons (12 core cpus min rec, 20 core preferred), min 32GB ram (usually at least 64,128+ common on high end systems), SSD for OS, thunderbolt (min)/pciE/10GbE/fibre (preferred) attached storage usually 8 bay raid6 or similar min, almost always NVIDIA GPUs with 8x 1080ti's or the latest Titans being the most common set up I see.

This runs on CentOS or RHEL 6.8 or 7.3.

Video signal is output over SDI from a PCIe to a LUT box (for color transforms) then to a color critical display (FSi, Sony, or Dolby typically with the best suites using cinema projectors). A second SDI runs out to a box showing video scopes. Everything is usually calibrated by light Illusions software and using a Minolta colorimeter probe (typically a 3rd party service does this every few months).

The GUI monitor(s) are usually just regular consumer whatever.

The software is controlled by a large, $30K control panel that looks similar to an airplane cockpit.

That's most of the important stuff, but I can fill in details where you're curious.

Are the ports on the 1080ti's used for video output at all? Is there one with SDI out? Or are they just used for CUDA?

At the risk of asking a silly question, what does the LUT-box do that couldn't be done in software (or, I guess, why isn't it done in software)?

This stuff is fascinating to me.

Do you know of any good YouTube videos on colorist hardware? I've seen a couple of videos on workflow, but neither went into the guts of the machines and LUT-boxes.

For 3D yes.