Where in the database is the password reset form's expiration date set? I'd like to be able to change the text in the temporary login page but I don't know where the expiration date for the form can be found. Getting the username is simple enough and I left the login button as is. The date is the only thing left.

Alternatively, can I do this in the drupal admin without having to code anything? Something similar to the email alerts, including options for multiple languages?
The text for the One time password form is located in user_pass_reset form function in core. 

While the user's password reset notification mail may be configured from the admin menu (Administer >> Configuration >> People >> Account Settings) there's no similar Admin UI based configuration of the reset form itself. 

Though there doesn't seem to be a module to customize the form itself (based on my search), there do appear to be modules that offer alternative password reset methods that you could look at namely Password Reset and Simple Password Reset.

If you could consider taking a coding way, this could be achieved by using a hook_form_alter or hook_form_FORM_ID_alter

As for the expiration date drupal uses a default of 86400 seconds (24 hours). It would be stored in the variable table in the database with the name user_password_reset_timeout (note that when present the value 86400 would be serialized like all values in that table). If you wish to change this value consider doing it using variable_set through code. The equivalent drush command would appear an easier option though. Do make sure to update the email content on expiry time to avoid any confusion.