What is the sharia's ruling about personal names found in newspapers etc that include either the prophet's name or Allah's name (Abdullah, AbdulKarim)? How may these papers be disposed of or destroyed?

Praise be to Allaah.
These papers on which Allaah’s name is mentioned should be kept and protected against being handled with disrespect until you have finished with them. When you have finished with them and no longer have any need of them, they should be buried in a clean place, or burnt, or kept in a place where they will be protected against disrespect, such as in a cupboard or on shelves, etc.

Fatwa of Shaykh Ibn Baaz from Fataawa Islamiyyah, vol. 4, p. 313.