What should I do about newspapers discarded in the streets that contain the name of Allah, or Qur'anic verses, or blessings upon the Prophet? Is it obligatory for me to inspect every newspaper, or is it sufficient to deal with what I see directly? And is ignoring distant newspapers considered an act of disbelief (kufr)?

It is not permissible to discard anything on which the name of Allah or a Qur'anic verse or a Prophetic Hadith (definition) is written. Whoever does so intentionally, with the purpose of mockery, becomes an apostate. Similarly, whoever finds a paper with a verse or Hadith on it and leaves it in an unclean place, thereby commits an act of disbelief. The determining factor is intention and purpose. So, whoever discards a page without intending to mock the small amount of Qur'an or Hadith on it does not commit an act of disbelief, but rather incurs sin. The matter should not reach the point of obsessive doubt and extremism. Anything found containing a verse or Hadith must be preserved, but if it is not highly probable that something is written on it, then there is no harm in leaving it.