How to add checkbox and radio button in Profile Page

You are missing the “checked” value for the inputs

<input type="checkbox" name="language" <?php if (get_the_author_meta( 'language', $user->ID) == 'Mandarin' ) { ?>checked="checked"<?php }?> value="Mandarin" /> Mandarin<br />

Also, the usermeta is dealing but your are checking for $_POST['gender']

Finally, you should have one usermeta for English and other for Mandarin, as they are not mutually exlcusive

[edit: working code]
[edit2: multiple languages]

<?php
$lingo = array('en' => 'English', 'md' => '普通話', 'es' => 'Español', 'fr' => 'Français', 'pt' => 'Português');

function my_user_field( $user ) {
    $gender = get_the_author_meta( 'dealing', $user->ID);
    $company = esc_attr( get_the_author_meta( 'company', $user->ID ) );
?>
    <h3><?php _e('More About You'); ?></h3>
    <table class="form-table">
        <tr>
            <th>
                <label for="Dealing Type"><?php _e('Gender'); ?>
            </label></th>
            <td><span class="description"><?php _e('Gender?'); ?></span><br>
            <label><input type="radio" name="dealing" <?php if ($gender == 'Male' ) { ?>checked="checked"<?php }?> value="Male">Male<br /></label>
            <label><input type="radio" name="dealing" <?php if ($gender == 'Female' ) { ?>checked="checked"<?php }?> value="Female">Female<br /></label>

            </td>
        </tr>
        <tr>
            <th>
                <label for="company"><?php _e('Company'); ?>
            </label></th>
            <td>
              <span class="description"><?php _e('Insert Your Company name'); ?></span><br>
                <input type="text" name="company" id="company" value="<?php echo $company; ?>" class="regular-text" /><br />
            </td>
        </tr>
        <tr>
            <th>
                <?php _e('Language'); ?>
            </th>
            <td><?php
                global $lingo;
                foreach($lingo as $key => $value) {
                    $code="language_".$key;
                    $lang = get_the_author_meta( $code, $user->ID);
                     ?>
                    <label><input type="checkbox" name="<?php echo $code; ?>" <?php if ($lang == 'yes' ) { ?>checked="checked"<?php }?> value="yes" /> <?php echo $value; ?></label><br />
                <?php }
            ?>
            </td>
        </tr>
    </table>
<?php 
}


function my_save_custom_user_profile_fields( $user_id ) {
    if ( !current_user_can( 'edit_user', $user_id ) )
        return FALSE;

    update_usermeta( $user_id, 'dealing', $_POST['dealing'] );
    update_usermeta( $user_id, 'company', $_POST['company'] );

    global $lingo;
    foreach($lingo as $key => $value) {
        $code = "language_".$key;
        update_usermeta( $user_id, $code, $_POST[$code] );
    }
}

add_action( 'show_user_profile', 'my_user_field' );
add_action( 'edit_user_profile', 'my_user_field' );
add_action( 'personal_options_update', 'my_save_custom_user_profile_fields' );
add_action( 'edit_user_profile_update', 'my_save_custom_user_profile_fields' );