Is it permissible for the man to go out wearing shorts?
It is permissible as long as his Awrah(private part), which is between his navel and knee, is covered. However, the regular wear traditions and modesty are to be observed.
Should I pick up the newspapers thrown at the street because they definitely contain sacred names?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
The Messenger of Allah (PBUH) says: "Be mindful of Allah and He will protect you." Part of this is protecting whatever contains the name of Allah. Therefore, if you see a paper containing Allah's name thrown in the street, then you should grab it and similar papers, then burn them in a clean place. Doing so is permissible because Othman (May Allah be pleased with him) has burnt copies of the Quran which didn`t match the main Mus-haf. However, you aren`t required to collect every paper in the street. And Allah The Almighty Knows Best.
Is it permissible to fast on behalf of the dead who died owing missed fast to make up ?
His relative should fast on his behalf, and it is permissible for the guardians to give permission to non-relatives of the dead to fast on his behalf as well.
Is it disliked to clip one`s nails at night, and is it permissible to burn them afterwards?
It is desirable to bury the cut hair and the clipped nails. As for burning clipped nails, there is no evidence forbidding that , but it is better to suffice with the aforesaid ruling-burying cut hair and clipped nails-as it is based on a well known evidence.