Unsolved homicides are never 'officially closed', as there is no statute of limitations on murder. Such a case would be classified as inactive.
Anyone working at the behest of the police is considered an agent and as such any evidence belonging to someone else (such as a family member) they seize from a home would be inadmissible in court.
Shell casings cannot be used to determine the manufacturer of a gun. The detective later comments about projectiles and lands and grooves, which can narrow down the list of manufacturers.
Contrary to what her father said, Alex did not leave the lights of her car on after she came home after spying on her father. While they were on when she pulled up, as she goes inside the left headlight is shown and it is not on. She did however leave her room lights on.
While evidence obtained and seized by Alex would not be admissible, information as to what she had seen could be used as probable cause in order to obtain a search warrant. As such, there would be no need for Alex to go back into the home. This was a large basis of the internal affairs charges shown at the end.