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 so 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
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.
What is the ruling on the chatting between a young man and his girlfriend?
Private chatting between the two sexes entails religious violations, and leads to prohibited attachment.
What should a praying person who doubts having offered one, or two prostrations in the third Rakah (unit of prayer) do?
Whoever has doubts about their prayer should consider what is less i.e. the above person should offer another Sajda (prostration) and complete his/her prayer, then offer Sujood As-Sahw (prostration of forgetfulness). And Allah Knows Best.