Your question mixes in several terms like focus and vignetting which I think show that you're making this more complicated for yourself than it really is.
Metering is just about determining the proper exposure for an image. It does this by measuring the light in a certain way, and computing an exposure value that matches the measurement. If you're in an automatic mode (P, Av, Tv, etc.), the camera will then adjust aperture, shutter speed, and/or ISO (all depending on the mode settings) to match the calculated exposure value.
The difference between spot metering and center-weighted metering is simply in what part of the scene is measured. With spot metering, it's only one very small part (almost always the center). Usually, this is about 4% of the frame, give or take a percent or two. Center metering takes the whole scene into consideration, but counts the broad middle with more weight. That means if there's something bright (like the sky) at the very top of your frame, it won't be factored in as much into the exposure calculation.
There is often a third type available, matrix metering, where the camera tries to match measurements from all around the scene to a database of likely scene types. That's outside of your question but important to think about because of the next thing I'm going to say:
None of this affects focus, which is a whole separate camera system. However, in some cameras (my Pentax K-7 is one, but other mid-range to advanced cameras will have similar features), the focus point selected can be used as data to influence the matrix metering, so that whatever you've selected to focus on is also given extra consideration in the calculation of exposure.
Finally, you mention vignetting. Vignetting doesn't come from metering or exposure changes except indirectly. There's three basic things that affect it.
First, something shading the actual edge of the frame. This could be a too-deep lens hood, or maybe a stack of filters.
Second, for whatever reason, you're using a lens that doesn't project an image circle big enough to cover the sensor. That's often the case with "toy camera" photographs.
And finally, a problem unique to digital. When shooting at large aperture, and particularly with wider-angle lenses, the light rays striking the sensor are completely perpendicular in the center of the frame, but at a skewed angle at the edges. The photosites in (most) digital sensors face straight ahead, and so the off-angle light is simply not recorded. This causes shading in the far edges and corner of the frame.
It is this third aspect which is linked to exposure, because a different exposure calculation can result in a larger aperture being chosen automatically, increasing vignetting. But it doesn't inherently matter which metering mode you've chosen.