Magento – How to export products with cross or up-sells products

You can build your $param array by merging the existing data with the products already assigned as crosssells.
You can get a list of the crosssels like this:

$crosssells = $product->getCrossSellProducts();

This will give you a collection with the products. To turn it int an array in the same form as $param you can do this

$param = array();
$crosssells = $product->getCrossSellProducts();
foreach ($crosssells as $item) {
    $param[$item->getId()] = array('position' => $item->getPosition());
}

Now you have in $param all the existing crosssels in a 'friendly' format.
All you need to do is to add your new crosssels:

if (!isset($param[$magentoID])){ //prevent elements from beeing overwritten
    $param[$magentoID]= array(
         'position'=>1
    )
}

After this just continue with your code:

$product->setCrossSellLinkData($param);
$product->save();

In Magento, cross-sell products are displayed in the shopping cart.
The reasoning is, if you place a product in the cart, they want to entice you to add one of the associated cross sell products, also.
For example, some nice wine glasses could be cross-sell products if you are buying a bottle of wine.
In a real life store, think of products at the end of the same isle.

The up-sell relation is to sell a better, newer, or generally more expensive variation of the product the customer is viewing. They are displayed on the product detail page.
For example, if you are buying the Heart Rate Monitor "newgen medicals 4in1" for 24.90 €, the "Wahoo Fitness Bluetooth Blue HR Smart 4031" for 69,90 € could be displayed as an up-sell product.
In a real life store, up-sell products would be placed right beside the other products, but probably a little higher on the shelf, maybe with a nicer display.

Note that this is how Magento defined these type of product associations. In contexts they might be used differently.