We can assume the error will have a random (whether it's actually truly random or merely pseudo-random doesn't matter here, just assume it's indistinguishable from truly random for this discussion) and a non-random component.
The random component I assume to be gaussian (thermal noise, for example) and therefore symmetrical around the real value.
It's obvious we can remove this type of noise through averaging (of which the core operation is integration).
The non-random component I assume to be a skew that can be calibrated out.
With these two assumptions in mind you can see that yes, it's indeed a random walk, but a very well behaved one.
No, you can't remove the random walk error by integrating. The point is that after integrating, what you're left with the random walk error. To make this concrete, if you buy a commercial-grade gyroscope for $10, it will have a random walk error of several º/√h. So after summing the errors for an hour, you're left with several degrees of random error, which is bad. If you spend $100,000 on a navigation-grade gyroscope, you'll get a random walk error < 0.002º/√h, which is much better.
As far as calibrating out the skew, of course you can do that to some extent, but it's not a magic bullet. The Minuteman periodically measures skew and even applies equations for the change in skew with acceleration. The problem is that skew is not constant; it changes with time, changes with temperature, changes with position, and changes randomly, so you can't just calibrate it out. That's one reason why missiles use strategic-grade IMUs for a million dollars rather than a commercial-grade gyro for $10: you're getting drift of .0001º/hour instead of .1º/second.
You are correct, I forgot to separate between long-term and short-term random effects.
Short-term random effects (as in, the part of the gyro's random walk error significantly higher in frequency than the inverse of the integration period) will get cancelled out by integration, assuming they're Gaussian.
Long-term random effects (mainly from time and temperature like you mentioned) will instead tend accumulate with integration aka worsening with time.
P.S. great fan of your many ventures into retro tech, keep them coming!
that was not what you forgot and your summary is still wrong. kens is correct. i suggest programming some simple simulations using a random number generator to get a better feel for the space
try it: take gaussian white noise with zero mean and integrate it twice. You'll see the signal does not stay close to zero, in fact it will drift arbitrarily far away from it over time (it's only necessary to integrate once for this to be true, but doing it twice as an IMU needs to will make it more obvious).
One can't cancel out random errors by integrating. You should take kragen's suggestion and write a quick simulation. To make this concrete, flip a coin 10 times. Take a step to the left for heads and a step to the right for tails. Most of the time you won't end up where you started, i.e. you have residual error.
The random component I assume to be gaussian (thermal noise, for example) and therefore symmetrical around the real value. It's obvious we can remove this type of noise through averaging (of which the core operation is integration).
The non-random component I assume to be a skew that can be calibrated out.
With these two assumptions in mind you can see that yes, it's indeed a random walk, but a very well behaved one.