So the big thing is theres alot of ways to interpolate (create the in between info) the information when you increase the number of lines. Some can be quite amazing, some terrible.
The most simple way is to obviously just double the size of the pixels to 2x2. This leaves really blocky looking images and is generally garbage.
The 2nd easiest way is to map the pixels to the bigger screen size at even spaces. Then simply average all the information between pixels to create a sort of smear effect as its blown up. It will be smoother than the doubling method, but quite blurry.
Now a good way to do it is it actually attempt edge detection. Since most objects tend to consist of a base color (or gradient) and an edge, you can attempt to keep the edges really sharp by drawing them precisely and not averaging the changes across the edges, then sort of smearing the flat color or gradient inside of the object. This is better than average because it reduces the blurring on edges, while still keeping the smoothness of surfaces.
On top of that theres many other methods to increase the number of lines, but basically the better the guessing algorithm the better the upconversion. One of the most well regarded upconverters is the HQV Reon chip, found in a few newer TV's and some newer Audio Receivers (Onkyo 875/905 for example).
So yeah, the reason an upconverting DVD player can improve the quality of a DVD is through the use of a high end upconverting chip running a more complex precise algorithm. The reason this is preferred, is depending on your brand of TV, the internal upconverting chip in the TV itself is often not as good as one found on the DVD player. Mostly because a TV has to play back the input in real time, but a DVD player can buffer (delay output) the audio and video and then send them simultaneously as the improved signal. Because a good upconversion algorithm can likely take longer than a simple one, the ability to buffer without hindering performance (games etc.) is pretty essential.
To answer your 2nd question. MAYBE... lol. It really depends on alot of factors including how good of a TV you have and how good of an upconverting DVD player you get.