If I have a 200mm f2.8 lens and apply a 2x teleconverter the effective focal length and aperture becomes 400mm and f5.6. However, I am unsure about the effect this has on other aspects, such as image sharpness (not of the teleconverter) related to the aperture. Say that my lens is very soft wide open at f2.8 and tack sharp when stopped down to f5.6. Will the added teleconverter give the same sharpness as 200mm f5.6 (minus the optical imperfections of the teleconverter and the effect of enlarging the lens imperfections) when shot at its widest (at f5.6), or will it be as soft as the 200mm f2.8 (plus the imperfections of the teleconverter)?
A TC merely magnifies what is projected out the rear of the lens. If the image your lens projects is soft at f/2.8, the TC behind it will magnify that softness.
If your lens is soft wide open at f/2.8 and considerably sharper at f/5.6, then you will need to stop the lens' aperture diaphragm down to the physical position it is in at f/5.6 when using only the bare lens to get the same improvement with the TC in place. This will give you an effective aperture of f/11. Most modern cameras will display the aperture setting as f/11 in such a case. But if your lens has an aperture ring that must be moved to stop down the lens, it will show f/5.6, even though the field density of light reaching your film or sensor will be equivalent to f/11 after the TC stretches the light circle projected by the bare lens.
To put it another way, your 200mm lens is wide open at f/2.8. When you add a TC, your 400mm lens is wide open at f/5.6. In both cases the actual aperture diaphragm is in the exact same position. You still need to stop down the physical aperture diaphragm two stops to increase the "sharpness" of the lens.
At best, the TC will always degrade image quality a little. Assuming you are using a good quality TC:
If your 200mm lens is soft at f/2.8, your effectively 400mm (lens + TC) will be slightly softer at effectively f/5.6. But that softness will be easier to see when magnified by 2X.
If your 200mm lens is sharper at f/5.6, your effectively 400mm (lens + TC) will be almost as sharp at effectively f/11. But the remaining imperfections in the improved image will still be easier to see when magnified 2X.
In that f/5.6 case, the optics will be slightly more degraded due to the addition of the teleconverter. You will very likely want to stop down to at least f/8 to use the teleconverter. Try all of these combinations, but expected practical results is that f/11 will be better yet with a teleconverter (but auto focus could become an issue then). It is not the same as a real 400 mm lens (but not real bad either).