The reason why teleconverters affect the effective aperture is that they are mounted behind the lens; they basically take all the light that enters the lens and spread it further outwards, so that the central portion now covers the entire sensor. The amount of light that is now aimed away from the sensor is "wasted".
Similarly, a wide angle converter takes the light that enters the lens, and points it inwards, so that part of the light that would normally end up besides the sensor, now lands up inside the sensor. This means there is more total light that ends up on the sensor, which means a faster effective aperture. Ofcourse if the lens is tailored for the sensor size, it won't have much light falling besides the sensor and a wide angle converter only creates vignetting, which is why the Speedbooster converters don't work on full frame cameras (unless they were to make a version that lets you use medium format lenses on full frame bodies).
Converters that are mounted in front of the lens (as opposed to between the lens and the body) usually don't change the effective aperture, since what matters to determine the f-stop is the size of the aperture as seen from the front of the lens (the direction from which the light is coming), and with a front-mounted converter, the aperture as "seen" by the incoming light changes with the same ratio as the angle of view.
As for DOF: that follows the light gathering ability every time; there's no separating light gathering ability and DOF where f-stops are concerned. That's because shallow DOF is caused by off-axis light (that doesn't travel in a straight line between whatever it's bouncing off of and the center of the lens) still gets captured by the outer regions of the lens, and then ends up somewhere near the straight-line light, but not at exactly the same place; in other words, the light from 1 point ends up in different places on the sensor, rendering that point blurry. Increasing the aperture size doesn't increase the amount of straight-line light that gets captured, but it does increase the amount of off-axis light.
If that's confusing, think of it as aiming pingpong balls at a small circle inside a bucket; increasing the bucket size doesn't increase the number of times you hit that small circle, but it does increase the number of pingpong balls that you manage to get in the bucket. The ones that would've missed the smaller bucket completely, now land inside the bucket, but somewhere besides the small circle. Those balls represent the result of a larger aperture; increasing the total light gathered, and blurring the image that's not on the focal plane.
Regarding the question of what delivers shallower DOF: it depends completely, based on which parameters you keep the same. A smaller sensor with the same absolute focal length and f-stop ( = same physical aperture size) will have shallower perceived depth of field, even though the absolute depth of field stays the same. The reason for this is that, assuming you view / print the images at the same size, the smaller sensor's image is magnified more than the bigger sensor's image. The result of that is that any blur that might be invisible due to the low magnification of the bigger sensor's image, does becomes visible in the smaller image. For instance, if the lens renders a point at 1m behind the subject as a 0.02mm blotch on the sensor plane, that would be 0.055% of the width of a full frame sensor, but 0.11% of a 4/3 sensor. If you print both images at the same size, the point will be spread out over a larger portion of the image with the smaller sensor, so the fact that it's rendered as a blotch instead of a point will become obvious sooner.
In other words, areas that might appear as sharply rendered points in a full frame sensor, can turn out to be blurry blotches in the 4/3 image, even though you used the same lens, f-stop and subject distance. That effect is expressed by the "circle of confusion" in the DOF calculator.
As for cropping affecting light-gathering ability: yes and no. The density of the light falling on the sensor doesn't change, but the total amount of light that you capture (or use) does change. If you then print / view the image at the same size as you would the uncropped image, you'll see more noise - since the noise that was there in the central portion of the image gets magnified - which makes the result the same as if you'd used a longer lens with slower f-stop and higher ISO on the uncropped image.