Couzens has pleaded guilty to the charges.
He sat in court with his head bowed as members of Ms Everard's family listened to prosecutor Tom Little open his case.
Mr Little said Couzens wore his police belt with handcuffs and used his police warrant card when he detained Ms Everard "by fraud" in a "false arrest".
He also had booked a car rental, the prosecutor argued.
There was "no credible alternative explanation for his need to hire a car other than to use that car to kidnap and rape a lone woman," Mr Little said.
"His movements were consistent with the defendant looking for, or hunting, for a lone young female to kidnap and rape, which is precisely what he did," the prosecutor argued.
Couzens joined the Metropolitan Police in 2018 and had worked as part of a team protecting diplomatic premises in central London.
He had worked an overnight shift at the US embassy on the day he kidnapped Ms Everard.
As a police officer, Couzens also had worked on COVID-19 patrols and enforcing coronavirus regulations, Mr Little said.
Ms Everard being out after going to a friend's house for dinner while the UK remained under lockdown made her more vulnerable to the officer's claim that she had breached pandemic rules, according to the prosecutor.
A passenger in a passing car witnessed the kidnapping but mistook it for an arrest by an undercover officer, he added.
Ms Everard's body was found in woodland in Ashford, Kent, about 97km southeast of London a week after she went missing.
Her disappearance led to one of the most widely publicised missing person investigations in the UK, Mr Little said.
The case also sparked outrage and triggered large-scale protests denouncing violence against women.
The UK government said after Ms Everard's killing that it would invest millions of pounds more in its "Safer Streets" fund to put more officers on the streets and improve street lighting and closed-circuit television facilities to protect women and girls.
Ahead of the court hearing, the Metropolitan Police department said it was "sickened, angered and devastated by this man's crimes, which betray everything we stand for".
The force said it would not further comment until the hearing is over.
A judge is expected to sentence Couzens on Thursday.
If you or someone you know is impacted by family and domestic violence or sexual assault, call 1800RESPECT on 1800 737 732 or visit 1800RESPECT.org.au. The Men’s Referral Service provides advice for men on domestic violence and can be contacted on 1300 766 491. In an emergency, call 000.