If an angled mirror is bolted to the equator, does the Earth change speed?

Yes, though the effect is minimal.

The radiation pressure $P$ on a perfectly reflecting mirror due to a beam of light hitting it perpendicularly is related to its intensity $I$ quite simply: $P = 2I/c$. For sunlight at 1 AU from the Sun (and ignoring the atmosphere for now), we have $I approx 1360 text{ W/m}^2$, so $P approx 9 text{ µPa}$. In other words, a mirror that was 1 km on each side ($10^6 text{ m}^2$) would experience a force of 9 N at most. We can see that whatever effect we're going to get will be quite small.

By tilting the mirror at an angle, we can get the reflected light to exert a torque about the center of the Earth. This will be dependent on the horizontal component of the radiation pressure experienced by the mirror. A bit of geometry shows that the horizontal component of the radiation pressure in this circumstance is reduced (for a fixed area) by a factor of $cos^2 theta sin theta$, where $theta$ is the angle between the sunlight and the normal to the mirror. Perhaps surprisingly, this is maximized not at $theta = pi/4$ but when $theta = cos^{-1}(sqrt{2/3}) approx 35.3°$. At this angle, the horizontal component of the radiation force will be $2/3sqrt{3} approx 0.384$ times the maximum radiation force, or about 3.5 N.

The atmosphere will reduce this number even further. The atmosphere absorbs about 23% of the solar radiation at 1 AU, reducing the amount of sunlight to 77% of the above. What's worse, the atmosphere will also absorb & scatter some of the sunlight on the way back out of the atmosphere, exerting a torque on the atmosphere opposite the torque on the Earth. Like the jet engine example you linked to, only the light that actually escapes the atmosphere will actually have a long-term effect on Earth's rotation. And since it's leaving at an angle relative to the vertical, its escape path through the atmosphere is longer, and so the atmosphere will absorb a larger fraction of the reflected light than of the incident light.

The maximum net effect is hard to calculate (in particular, it'll change the optimum angle I found above). But if I had to estimate, I would expect that you couldn't get more than 2 N of tangential thrust out of the radiation pressure from the Sun on a 1-km-square mirror on Earth's surface. It is left as an exercise to the reader to figure out how quickly this will change the rate of rotation of the Earth.

Yes, with the proviso that such effects are below the threshold of detectability and smaller than other effects that are not measured.

Any unbalanced external torque will affect an object's instantaneous angular momentum. In the case of the earth this might include:

• Lunar (and other solar system objects) gravitational forces on the non-spherical earth
• Solar wind interaction with the earth's atmosphere. Very possible that the temperature differences at the morning and evening terminators create an asymmetry that would allow a net torque.
• Asymmetric distribution of inbound material such as micrometeorites
• Rocket launched material that that does not return to earth in the timeframe under consideration
• Light pressure from your mirror

It's similar to saying that the exhaling breath from a batter at the plate changes the path taken by a pitched baseball. Since the presence and motion of the air affects the ball, this is technically true. But the size of the force and the amount of time that it has to act make the change too small to detect.