It is permissible to keep cats in the house, and there is nothing wrong with that because cats are not harmful or impure. For more, see the detailed answer.

Ostrich meat is halal because Allah has blessed His slaves by subjugating to them that which is in the heavens and on earth. For more about haram animal meat in Islam, see the detailed answer.

Trimming the beard is haram as the Sunnah of the Prophet (peace and blessings be upon him) indicates that it is obligatory to let the beard grow and that it is haram to shave it or cut it.

The papers which have Allah’s name can be put to good use by giving them to people who can benefit from them, or placing them in a library for that purpose. If that is not possible, then they may be destroyed by burning them, burying them, or shredding them so that the verses of the Quran and the name of Allah disappear.