How big (in meters) could the difference be between ECI and ECEF coordinates at midnight UTC? Is the only difference due to leap seconds, etc. or other differences between UTC and updated forms of universal time, such as UT1?  In other words, are all earth-centered inertial (ECI) coordinate frames constructed on purpose so they match at approximately midnight each day, and if so, how big of a difference could there be in either the X, Y, or Z coordinates (in meters)?
 A: 
In other words, are all earth-centered inertial (ECI) coordinate frames constructed on purpose so they match at approximately midnight each day?

No, they are not.
There are multiple Earth centered inertial frames. All have one thing in common: An Earth-Centered, Earth-Fixed rotates more or less about the ECEF frame's z axis at the rate of more or less one rotation per sidereal day -- not one rotation per mean solar day. Your question implicitly assumes that the Earth rotates once per mean solar day. Ignoring the equation of time, the Earth does indeed rotate once per mean solar day with respect to the Sun. All Earth-centered inertial frames are defined in terms of the "fixed stars" rather than with respect to the Sun.
Suppose you asked about ECI coordinates of a point with fixed ECEF coordinates one sidereal day later rather than one mean solar day later. Note that I used "more or less" twice. The Earth's instantaneous rotation axis is not quite the same as the ECEF z axis. The axis of revolution moves around a bit with respect to the Earth's crust; this is called polar motion. The other "more or less" was the rotation rate. This, too, varies a bit. (It is also slowing down, but this is very gradual.)
One final issue is the orientation of the ECEF z axis (ignoring polar motion) and the ECI z axis. In addition to rotation and polar motion, the Earth also undergoes precession and nutation. Precession is slow but rather large. The ECI coordinates of a point with fixed ECEF coordinates change a bit over exactly one sidereal day, and change a lot after exactly 100000 sidereal days.
