To make the distinction: PAL is just an encoding system, which only really defines the resolution and frame rate for analogue systems. Rec601/709 is a colourspaceare colourspaces generally used by TVs.
Sounds like you might be having trouble with colourspace conversion. The black point on an 8-bit Rec601 image is 16, not 0 (it's to do with the way analogue systems transmit video data - sync, etc.), so as a result, anything below this, from 0-15 will be displayed as black.
Video editors, DVD burners and other video processors may not take colourspaces into account or may manipulate your video, resulting in a change of colourspace, which will ultimately change the colours in your video.
The 'Sony Levels' effect essentially applies a LUT to your image (think of it as scaling the values 0-255 to the bounds 15-235, the 8-bit luma range). This means that your image will be displayed correctly on a Rec601 system, but will look flat on a computer.
Working in a 10+ bit workflow may help to solve this issue. However, to answer your points:
1. Is this the correct way to handle the situation?
Your camera shoots video in the RGB colourspace, not the Rec601/709 spaces.
Ultimately, the best way is to apply a RGB -> Rec601 LUT, essentially remapping the colours correctly across the board, but I'm not sure if Vegas supports this off the top of my head. Using the Levels effect will help with the black/white values (make sure the centre is at 0.5), but there may be some colour variation.
2. What is the correct value to use for the levels effect?
The black point should be 0.059 and the white point should be 0.918 to comply with the standard.
3. Is it correct to not apply the levels effect when rendering for Youtube etc.?
That's correct - YouTube will accept your video with the default colourspace and should display it correctly. You may as well test this, though.
4. In which situations do I have to use the levels effect, and in which situations better not?
Any time you're displaying your content on a Rec601 device.