Is it permissible for the heirs of the person who caused the accident (The Killer) to fast (The intended as an expiation for unintentional killing) on his behalf if the latter died in the crash?
                                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.
 
One who dies while liable for obligatory fast, his/her guardians have the choice to either fast on his/her behalf or feed one poor Muslim for each day of missed fast. This applies to Ramadan and other obligatory fast, and since expiatory fast is obligatory, feeding a poor person for each missed day or fasting on behalf of the deceased is obligatory as well. And Allah The Almighty Knows Best.
                             
                        
                            
                                
                                What is the ruling in Islam on castrating animals to use them for work? 
                                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.
This isn`t permissible because jurists said: "It is permissible to castrate the animal if it is eaten and this procedure is done at early age." And Allah The Almighty Knows Best.
                             
                        
                            
                                
                                Is it permissible to swear by (Allah "The Living")?
                                Yes, it is permissible because "The Living" is one of Allah`s Attributes.
                             
                        
                            
                                
                                What is the ruling of Sharia on charging one`s mobile from the plug of the university`s lecture room?
                                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 pious wouldn`t do that. And Allah The Almighty Knows Best.