Magento 2 – Schema Changes for EE Catalog Staging

The reason is that you're saving the product but you've not loaded the visibility as part of the product collection. Catalog, Search is the default visibility.

Try this:

$productCollection = Mage::getModel('catalog/category')->load($categoryId)
    ->getProductCollection() 
    ->addAttributeToSelect(array('name','price','special_price','sku','small_image'))
    ->joinAttribute('visibility', 'catalog_product/visibility', 'entity_id', null, 'inner');

Now when you save the product it should retain its original visibility setting.

Place InstallData.php file in your module's Setup directory:

<?php

namespace Training\Product\Setup;

use Magento\Framework\Setup\InstallDataInterface;
use Magento\Framework\Setup\ModuleContextInterface;
use Magento\Framework\Setup\ModuleDataSetupInterface;
use Magento\Catalog\Setup\CategorySetupFactory;

/**
 * @codeCoverageIgnore
 */
class InstallData implements InstallDataInterface
{
    /**
     * Category setup factory
     *
     * @var CategorySetupFactory
     */
    private $categorySetupFactory;

    /**
     * Init
     *
     * @param CategorySetupFactory $categorySetupFactory
     */
    public function __construct(CategorySetupFactory $categorySetupFactory)
    {
        $this->categorySetupFactory = $categorySetupFactory;
    }

    /**
     * {@inheritdoc}
     */
    public function install(ModuleDataSetupInterface $setup, ModuleContextInterface $context)
    {
        $setup->startSetup();

        $categorySetup = $this->categorySetupFactory->create(['setup' => $setup]);

        $categorySetup->addAttribute(
            \Magento\Catalog\Model\Product::ENTITY,
            'some_new_attr',
            [
                'label'            => 'Some Cool Attribute',
                'group'            => 'Product Details',
                'required'         => false,
                'visible_on_front' => true,
            ]
        );

        $setup->endSetup();
    }
}

Then add your module to app/etc/config.php, clear cache and install:

bin/magento setup:upgrade

This should be enough, no need to add anything in XML files. This should answer your second question.

Regarding first one, data from Magento\Catalog\Setup\CategorySetup::getDefaultEntities() is used by Magento\Eav\Setup\EavSetup::installEntities() method.