I would reinstall WP (don’t overwrite your wp-config.php file). Then reinstall all themes/plugins from original source. You may have to do this with FTP if you don’t have access to admin area. Although FTP is slower (depends on your connection speed), that will ensure that all files have latest versions.
Then, use File Manager to look at all folders, sorting by date, looking for files that don’t match the datestamp of the other files (since you updated everything). Look for files that don’t belong. Could be a hack attempt.
Test the mail() function by creating a simple PHP page with a valid mail() command. Make sure that your admin email account is of the same domain as the site. If the admin account is a different domain name than your site, mail messages will be not sent by your server or caught as spam.