What do I do when an extension overwrites a class globally and I want to use the original?

Caveats: There's no designed way to do what you're asking in the system. The following should work, but I've never tried it out extensively on a production system, and there may be situations where it will cause more trouble that it's worth. Only proceed if you're comfortable debugging problems related to the changing the rewrites of a working system.

Step 1 is undoing the rewrite. The Magento configuration tree can be changed at runtime. So, if you run the following code

$config = Mage::getConfig();        
$config->setNode(
    'global/blocks/catalog/rewrite/product_list_toolbar',
    'Mage_Catalog_Block_Product_List_Toolbar'
);

Then Magento will instantiate the original Mage_Catalog_Block_Product_List_Toolbar block for the remainder of the request.

Step 2 is deciding where to call this in your module. Since this is just for your controller and it's rewriting a block that won't be instantiated until the end of your controller, I'd add a method to your controller class something like this

protected function _undoRewrites()
{
    $config = Mage::getConfig();        
    $config->setNode(
        'global/blocks/catalog/rewrite/product_list_toolbar',
        'Mage_Catalog_Block_Product_List_Toolbar'
    );    
}

and then just call this method at the start of each of your actions

public function indexAction()
{
    $this->_undoRewrites();
    $test = Mage::getSingleton('core/layout')->createBlock('catalog/product_list_toolbar');        
    var_dump($test);
}

This may seem a little clunky, but I think it's a good idea to be clunky (i.e. obvious) when you're being clever with Magento's system objects. Another place for this could be the controller_action_predispatch or controller_action_predispatch_front_controller_action events and/or applied conditionally.

Just remember the rewrite won't be undone until this method is called. That means if you attempt to instantiate a block before calling _undoRewrites, the rewritten class will be used to instantiate the object.

Solution 1:
You can try to instantiate the class directly (php way) in your controller

instead of

$this->getLayout()->createBlock('catalog/product_list_toolbar');

something like:

$block = New Magento_Catalog_Product_List_Toolbar;
$this->getLayout()->addBlock(....);

Solution 2:
Another approach would be create a new class, in your module, that extends the original class and use that one.

Solution 3:
Otherwise if the extension is not crypted ( we all love open source :) you can try to find out why it breaks your stuff

If multiple rewrites exist for the same class alias, then the last one the Magento config loader parses from config.xml "wins". I'd attack this problem by:

1. Create a new extension of your own.
2. Rewrite the catalog/product_list_toolbar in your extension
3. Have your block extend Mage_Catalog_Block_Product_List_Toolbar instead of the Amasty class.
4. Liberally comment your class explaining that this rewrite conflict is intentional. You don't want another developer who runs MageRun to try and "fix" the rewrite conflict you've just created.
5. Add a dependency in your extension's app/etc/modules/blah.xml file to ensure your extension is loaded after the Amasty one.