- Does the basil on the left side have any chances of survival? It doesn't seem likely, but there is still some green on the leaves.
The plant on the left side is too far gone. It will never fully recover, and should be disposed of. The plant looks like it either took some cold, or got basil fusarium wilt. I think the latter is unlikely, but I'd keep that plant away from the other, just in case.
- The one from the right has some leaves with black spots. Should I remove them entirely or only the black spots?
This is probably cold damage. There is a slight chance that the spots are infected, so to be safe, remove the leaves with black spots, or if the spotting is all at the end, you can cut the spots off and leave a portion of leaf on the plant.
- I live on the second floor of a big building. My apartment has windows facing east and west. Is it recommended to move the plan during the day in order to have more sun, or is it recommended to leave it in the same place?
Basil likes as much sun as it can get, especially indoors (and even more especially in Poland). It will benefit the plants if you can move them to whichever window has the most light at the time, but be gentle.
Your plant looks fairly healthy, but there are some good points to know, for a beginner trying to grow herbs indoors.
Do not overwater. This is probably the number one reason indoor herbs fail. Indoors, the plants don't grow as fast, and the potting mix doesn't dry out as fast. water only when the top layer of mix dries out, becoming several shades lighter in color.
Provide as much light as possible. Move to the sunniest windows whenever possible.
Gas ranges can put fumes into the air which will kill plants. Be aware that even if you can't smell the fumes, they can still kill plants. If you have a gas range, keep the plants as far from it as possible.
Basil likes humidity, so spraying with pure water now and then will benefit the plant, while cleaning off the accumulated dust. Air conditioning will cause a loss of vigor, and maybe eventual death.
The temperature matters to basil. The warmer the better, indoors. As long as it stays above 70 degrees Fahrenheit, they should be fine.
Below are links to more information on this subject which may be helpful in your case:
Starting growing herbs inside, is there anything I should worry about?
How do I help my kitchen window herb garden thrive?
How long will my basil plants live inside?
How can I grow herbs in my kitchen?