Is it sufficient to perform ghusl with the intention and by making water flow over the entire body, even if impurities are not removed before or during the ghusl? And if the hand becomes contaminated while washing the private parts, is it sufficient for purification to cleanse the spot touched by the hand and then complete the ghusl without paying further attention to that?

It appears the questioner is afflicted with purity obsessive-compulsive disorder (waswasat al-tahara). The Shari'ah has warned against succumbing to obsessions due to the hardship and difficulty they cause. Rather, a person afflicted with obsessions must strive against himself to be rid of them. He must know that Allah is kind to His servants, and His law is tolerant and easy. If a Muslim covers his body with water, his ghusl (ritual bath) is complete. If there is a certainty of impurity, its location must be purified, but its persistence does not affect the validity of the ghusl.