With underexposure, a third of a stop won't hurt you much. Two-thirds is not great, but not that bad. A whole stop means you're doubling your noise. Two stops and you're quadrupling it. So underexposure is by no means "free", but there are grades of it.
With overexposure, any amount will start to clip. Whether that is very detrimental will depend on the subject matter, but it's generally more damaging to your image than the noise of underexposure.
Now to your questions.
- What should I do? Should I do it that way, or does it depend on the circumstances (like if I have enough time to test for proper exposure, etc.). Or should I rather try to learn to "guess" the right settings (I will fail a lot but also learn a lot)?
No, don't underexpose. Aim to expose correctly. If you are not sure if you've exposed correctly, then it's better to underexpose a little than overexpose a little, that is true, though it still won't get results as good as exposing correctly. If you have the opportunity to try again if you make a mistake, it's better to aim for a correct exposure and try again later if you don't get it.
- What do you do? How do you aproach a shoot?
I try to get it spot on. I check the display on the LCD, and might check the RGB histogram, and if I've under- or over- exposed then I take another one. Of course this is not a luxury you have if you're doing photojournalism or anything like that, where you can't re-try the shot.
- How does underexposing affect dynamic range of photos? Does it effectively reduce contrast?
Underexposing reduces dynamic range by removing some range in the darkest areas, such as shadows. So underexposing by one stop reduces your dynamic range by one stop (ie, by half). The noise floor raises, so the loss in dynamic range is expressed as an increase in noise in the darkest parts of the image, obscuring some of the range that otherwise would have shown dark shadow detail.
I wouldn't say that it reduces "contrast" as such. It does reduce the dynamic range - that is, the relative distance between the darkest and lightest detail you can reproduce without it being obscured by noise. But I wouldn't describe that as primarily an effect on contrast - depending how you use the word "contrast".