Give error if checkbox isn't selected on Woocommerce Account registration

Question

I'm trying to add some custom fields in the Woocommerce User account registration form.
I've succesfully added First Name and last name, nut i'm trying to add a T&C agreement checkbox, that if it isn't selected it doesn't let the user continue the registration.
I've managed to display it correctly the i cannot make the validation work. What am i doing wrong?

PS. just to know, is it possibile to make these checks live, without needing to reload the page?
Thanks
The code i've used:

add_action( 'woocommerce_register_form_start', 'cbi_custom_woo_account_registration' );

function cbi_custom_woo_account_registration() {
    ?>

    <p class="woocommerce-FormRow woocommerce-FormRow--wide form-row form-row-wide first">
    <label for="reg_billing_first_name"><?php _e( 'First name', 'custom-cbi-parts' ); ?> <span class="required">*</span></label>
    <input type="text" class="woocommerce-Input woocommerce-Input--text input-text form-control" name="billing_first_name" id="reg_billing_first_name" value="<?php if ( ! empty( $_POST['billing_first_name'] ) ) esc_attr_e( $_POST['billing_first_name'] ); ?>" />
    </p>

    <p class="woocommerce-FormRow woocommerce-FormRow--wide form-row form-row-wide">
    <label for="reg_billing_last_name"><?php _e( 'Last name', 'custom-cbi-parts' ); ?> <span class="required">*</span></label>
    <input type="text" class="woocommerce-Input woocommerce-Input--text input-text form-control" name="billing_last_name" id="reg_billing_last_name" value="<?php if ( ! empty( $_POST['billing_last_name'] ) ) esc_attr_e( $_POST['billing_last_name'] ); ?>" />
    </p>

    <p class="woocommerce-FormRow woocommerce-FormRow--wide form-row form-row-wide">
    <label for="accept_terms"><?php _e( 'I agree TeC', 'custom-cbi-parts' ); ?> <span class="required">*</span></label>
    <input type="checkbox" class="woocommerce-Input woocommerce-Input--text input-text form-control" name="accept_terms" id="accept_terms" value="<?php if ( ! empty( $_POST['accept_terms'] ) ) esc_attr_e( $_POST['accept_terms'] ); ?>" />
    </p>

    <div class="clear"></div>

    <?php
}

// 2. VALIDATE FIELDS

add_filter( 'woocommerce_registration_errors', 'cbi_validate_custom_reg_fields', 10, 3 );

function cbi_validate_custom_reg_fields( $errors, $username, $email ) {
    if ( isset( $_POST['billing_first_name'] ) && empty( $_POST['billing_first_name'] ) ) {
        $errors->add( 'billing_first_name_error', __( 'First name is required!', 'custom-cbi-parts' ) );
    }
    if ( isset( $_POST['billing_last_name'] ) && empty( $_POST['billing_last_name'] ) ) {
        $errors->add( 'billing_last_name_error', __( 'Last name is required!.', 'custom-cbi-parts' ) );
    }
    if ( isset($_POST['accept_terms']) && $_POST['accept_terms'] == 0 ) {
        $errors->add( 'accept_terms_error', __( 'You must accept Terms and Conditions!.', 'custom-cbi-parts' ) );
    }
    return $errors;
}

Answer 1

I have fixed your code.

add_action( 'woocommerce_register_form_start', 'cbi_custom_woo_account_registration' );

function cbi_custom_woo_account_registration() {
    ?>

    <p class="woocommerce-FormRow woocommerce-FormRow--wide form-row form-row-wide first">
    <label for="reg_billing_first_name"><?php _e( 'First name', 'custom-cbi-parts' ); ?> <span class="required">*</span></label>
    <input type="text" class="woocommerce-Input woocommerce-Input--text input-text form-control" name="billing_first_name" id="reg_billing_first_name" value="<?php if ( ! empty( $_POST['billing_first_name'] ) ) esc_attr_e( $_POST['billing_first_name'] ); ?>" />
    </p>

    <p class="woocommerce-FormRow woocommerce-FormRow--wide form-row form-row-wide">
    <label for="reg_billing_last_name"><?php _e( 'Last name', 'custom-cbi-parts' ); ?> <span class="required">*</span></label>
    <input type="text" class="woocommerce-Input woocommerce-Input--text input-text form-control" name="billing_last_name" id="reg_billing_last_name" value="<?php if ( ! empty( $_POST['billing_last_name'] ) ) esc_attr_e( $_POST['billing_last_name'] ); ?>" />
    </p>

    <p class="woocommerce-FormRow woocommerce-FormRow--wide form-row form-row-wide">
    <label for="accept_terms"><?php _e( 'I agree TeC', 'custom-cbi-parts' ); ?> <span class="required">*</span></label>
    <input type="checkbox" class="woocommerce-Input woocommerce-Input--text input-text form-control" name="accept_terms" id="accept_terms" <?php if( isset($_POST['accept_terms'])){ echo "checked"; } ?> />
    </p>

    <div class="clear"></div>

    <?php
}

// 2. VALIDATE FIELDS

add_action( 'woocommerce_registration_errors', 'cbi_validate_custom_reg_fields', 10, 3 );

function cbi_validate_custom_reg_fields( $errors, $username, $email ) {

    if ( isset( $_POST['billing_first_name'] ) && empty( $_POST['billing_first_name'] ) ) {
        $errors->add( 'billing_first_name_error', __( 'First name is required!', 'custom-cbi-parts' ) );
    }
    if ( isset( $_POST['billing_last_name'] ) && empty( $_POST['billing_last_name'] ) ) {
        $errors->add( 'billing_last_name_error', __( 'Last name is required!.', 'custom-cbi-parts' ) );
    }
    if ( !isset($_POST['accept_terms']) ) {
        $errors->add( 'accept_terms_error', __( 'You must accept Terms and Conditions!.', 'custom-cbi-parts' ) );
    }
    return $errors;
}