Horizontal lines are usually due to power fluctuations causing brightness differences. Since your monitor's scan rate is 60 Hz while the power line frequency is also 60 Hz, a very small deviation in either from 60 Hz causes the line to slowly crawl up or down.
Try plugging the computer + monitor into a power strip (or a higher quality one if you're using a cheap one), or different outlets. If the power in your area is just poor quality, there's not much you can do about it. If you've got a UPS you can try plugging into that, but those usually just pass through the wall outlet power until they detect a power failure.
As for your screen being bright even though you're displaying black, that's an inherent property of the monitor. Some monitors have a bright black point, others (more expensive ones) a dark black points. That combined with the white point (brightest white) determine the contrast ratio. So you can usually avoid this problem if you check reviews for the monitor's contrast ratio (not dynamic contrast ratio - that's a meaningless marketing number). Anything above 1000:1 is usually pretty good. The poor ones are usually down around 300:1. Anything above 3000:1 is referring to dynamic contrast ratio and should be ignored.
The best are plasma and OLED displays, which emit no light at all when displaying black. So their black points give off no light, and they have an infinite contrast ratio. It used to be IPS-type displays were better at this than cheap TN panels. But the TN panels have gotten much better, while I've noticed LG has been trying to make its IPS screens cheaper causing a brighter black point, especially when viewed from the sides or diagonally.
Since you're stuck with your monitor, the only thing you can do is try to turn down the backlight brightness (which confusingly is controlled by the contrast setting). Unfortunately that will also make your white point darker (there's no escaping contrast ratio). So you will just have to find a balance between bright blacks and dark whites that you can live with.