What are the grounds for a lawful vehicle search after a traffic stop?

Question

Description

Read the scenario and answer the 4 questions that follow:

Officer Bird was patrolling downtown Center City late Saturday night. Primarily, she was on the lookout for drunk drivers after the bars closed, but violent incidents were on the rise in the area, so she was also looking for suspicious activity. She had just pulled over across the street from the Central Pub when a red sports car went speeding by. Officer Bird turned on her siren and pursued the sports car. The car pulled over, and Officer Bird approached the vehicle. When she got to the door, she noticed the driver shoving something under the front seat. She rapped on the window, and the driver, visibly angry, rolled it down.
“What?” he said.
 Officer Bird told him to step out of the vehicle. The driver slammed the door and stepped toward Officer Bird. She then ordered him to turn around and place his hands on the roof of the vehicle. The man swore, but he complied. Officer Bird did a weapons pat-down and did not find a weapon.
“OK, Sir, now I need to see some identification.” When the man handed over his license, she saw that his name was Alejandro Quintana. Before running his license, she shined her flashlight in the sports car. Sticking out from under the front seat, she saw a revolver. She seized it. In addition, she also saw a small bag containing a white powdery substance next to the revolver. She seized that as well.
“You should see what’s in the trunk,” Quintana said menacingly.
 Officer Bird just placed Quintana under arrest, and did not check out the trunk.

What are the grounds for a lawful vehicle search after a traffic stop?

What Supreme Court decision(s) governs this activity?

Did Officer Bird’s actions (search and arrest) satisfy the constitutional requirement(s)?

Could Officer Bird have legally searched the trunk? Why or why not?