I am hoping that by adding a telecompressor, I will be able to reduce the focal length, thus increasing the image resolution/quality captured by the camera.
1) Is it correct to assume that by using a telecompressor, I reduce the focal length, and thus increase the quality of the image because I am getting the image closer to the camera?
2) what are the drawbacks of using the telecompressor?
Adding optical elements that are not part of the original lens design almost always introduces additional optical errors due to some degree of "mismatch." Image quality is (almost?) always reduced to some degree (but the benefits can outweigh that negative).
Reducing the magnification/increasing the FOV can increase apparent image quality by reducing the apparent movement (camera/subject). And increasing the FOV can increase image quality by requiring the camera to be closer to the subject (for a given composition), which makes the details larger/clearer to start with. But those two things counter each other... moving closer inherently increases the effective magnification.
Changing the focal length does not change image quality but the view angle. That is, it changes how much you see of your object or how large you see things and how much of their surroundings.
Introducing an additional lens, not designed specifically for your existing lens, does not improve colours or sharpness either, if there are chromatic errors like aberration or others like koma. On the contrary, additional optics usually add further small errors, unless shaped to specifically improve on known deficiencies (like the glasses one might wear). A focal reducer is not one of these optical-error-correcting things and one does not use them unless external circumstance dictate you a viewing distance and you cannot see everything you need to see.
It may sometimes seem that they do improve image quality - when your optics are already operating beyond the diffraction limit and your existing magnification is actually too large for the apperture you have. Then reducing the magnification will not hurt, but on contrary will limit the diffraction to about pixel size or less and simply give you a wider FOV.
Apart from the other answers stressing the addition of optical elements, it's worth pointing out that a wide extender (never heard the term telecompressor) will change the relation between camera movement and image movement. If the camera does not have a specific option for that case, this means that image stabilisation will in general be overcompensating significantly.
Coming back to the additional optical elements, this can in rare cases be an advantage: a zoom lens does not have option to remove elements at one end of the range that have action not suitable for that end of the range. It can only change their distance around. Being able to add additional elements, particularly at the extreme wide end, can be an advantage for the performance at the other end of the range which does not have to contend with having to compensate for those elements.
However, it will usually be extenders specifically made for a particular camera/lens model that will perform best. Generic components have no way of knowing which particular optical defects of the main lens at the extender's target range may be most worthy of compensating to what degree.
The short answer is 'no'. It does the opposite. But you may be able to play the degradation to your advantage.
Why?
Image field 'flatness' with respect to focus
Lenses project an image onto a sensor. If you focus the lens at some target in the center of the frame (image center has the best focus the lens can offer) and compare this to the quality you get near the edges and corners of the frame, you'll notice there is a slight degradation of focus as you get farther from the center of frame. BTW, you can adjust focus until you find the corner is optimally focused, but then you'll notice the center is no longer optimally focused.
This happens because lenses do not have a "flat" plane of focus. It is slightly curved. Corrective elements are used to improve the flatness, but this just makes the lens "nearly" flat... not "perfectly" flat.
Image field 'flatness' with respect to optical aberrations
There are numerous optical aberrations depending on the optical train. Coma is common. There are other types of aberrations but it's not easy to generalize them.
These aberrations mean that if you had a point-source of light, it would no longer focus as a "point" ... it might be a larger "spot" instead of a point. But it will often change shape entirely... and become elongated or start to form other interesting shapes (tear-drop shapes, or bat-wing shapes, etc. these vary depending on the type and severity of the aberration and that will be based on the optical train.)
Effects of increasing the angle of view
A lens doesn't project a neat rectangular image... it projects a large spot into the camera body ... but much of the light falls outside the boundaries of the sensor.
The image degradation I mentioned above continue to degrade as you inspect areas of the image that are farther away from the image center (presuming you optimized focus for the center). This means those areas outside the boundaries of the sensor are even more degraded than the corners.
When you use a focal reducer (tele-compressor) to reduce the focal length, you increase the lenses true angle of view and parts of the image that would have been outside the frame will now appear as part of your image.
The typical effect is that you get a wider angle of view, but the edges and corners are suffer from even more degradation of focus and optical aberrations than were present without the tele-compressor.
That aside... sometimes the optical aberrations are taken as an artistic element in the image (for example a "swirled" bokeh effect). The lens wont be capable of delivering a tack sharp image from corner to corner ... but if you play the effect nicely, you might enjoy the images you can capture with it.
