On local machine can’t log in or reset password but I can log in on the live version, user has activation key in database

Your local server probably has problems sending mail. There are ways to fix that, but it is probably easiest to change the password in the database directly.
Since you’ve made a copy of your site I am going to assume you have access to the database through something like PhpMyAdmin.
Follow these instructions to change the password: https://codex.wordpress.org/Resetting_Your_Password#Through_phpMyAdmin
Short version: find your user in the wp_users table, type a new password in the user_pass column and make sure to select MD5 under functions.

To prevent this problem in the future you could install a SMTP mail plugin. There are quite a few in the plugin directory for this.

Good luck!

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