Greetings,
I have excellent skills and experience in wordpress and php.
Have worked on projects similar to your requirements.
For users to login on the separate php file using their existing wordpress usernames and password,
you will need to connect a database to that file containing the wordpress login details.
The best option for the database would be MySQL.
I have done work like this before and easily do it for you in the given timeline.
Hoping to hear from you,
Regards,
Rishabh