For account moderation (allow login only after email verification), there are lots of plugin, you can try Theme My Login, it has moderation option.
For using your email address as sender, there are two ways, if you want to send VIA SMTP server you can use any SMTP plugin like Easy WP SMTP or any other as per your preference or if you don’t want to use SMTP, you can mask the sender name and email by copying the following to functions.php
// Change email address
function my_sender_email( $original_email_address ) {
return '[email protected]';
}
// Change sender name
function my_sender_name( $original_email_from ) {
return 'John Doe';
}
// Hook to WordPress filters
add_filter( 'wp_mail_from', 'my_sender_email' );
add_filter( 'wp_mail_from_name', 'my_sender_name' );