Is Iddah (waiting period) incumbent on an old lady whose husband had died?

Upon the death of the husband, Iddah is incumbent on the wife regardless of her age.

Is it permissible for the man to go out wearing shorts?

All Perfect Praise is due to Allah, The Lord of The Worlds

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. And Allah Knows Best

Is it permissible for a woman to swear a false oath by the Holy Quran to please her distrustful husband?

It is forbidden for a person to do so since this is considered a false oath(Yamaan Ghamoos) which dips(Yaghmos) its taker in Hellfire if he/she doesn`t seek repentance, and he/she has to expiate for that oath. You should avoid any doubtful acts so that your husband`s chest remains clear from the evil suggestions of the devil.

Should vowed fast be offered in consecutive days, or not?

Vowed fast should be offered in line with what was originally intended by the vow-maker, either consecutively, or not.