Question: I have some skin cracks around my nails (i.e. some parts of skin don't fasten tightly to the body, so as I understand I should wash above and under this part of skin), and it's always a problem to wash them: by just pouring water, I'm not sure all my skin will get that, but by rubbing every bit of skin with fingers - it's too long.
Answer: If the skin is loose, then you must make sure that the water reaches, unless this harms you. I haven't seen your skin cracks, but it seems you are exaggerating by rubbing every bit individually. Water would normally penetrate cracks, especially if you rub the area of the cracks (not every crack individually) with flowing water.
Question: Also near the roots of nails there are some "small nails" (i think everybody has them): the problem is that they are not tightly fastened to my nails - that has appeared recently (because I touched them too often): by just poring water, I'm absolutely unsure that the area under these little nails (but it's over real nails) will get wet (as it's smth. like skin cracks, I think that I should wet the area under these "little nails" also), what should I do?
See above answer.
Question: During wudhu or ghusl: if after washing the face, I scrape, and some piece of skin or hair fell out, should I wash this part again (because this "new appeared" area (under skin, hair) hasn't been washed)?
Answer: It is not necessary to do that.