Magento – How to get the quote from the admin session

The general idea is to pass the scope, chosen, through the request parameter.

I think you should use the similar code like in the product edit action:

/** @var \Magento\Store\Model\StoreManagerInterface $storeManager */
$storeManager = $this->_objectManager->get('Magento\Store\Model\StoreManagerInterface');
$storeId = (int) $this->getRequest()->getParam('store', 0);
$store = $storeManager->getStore($storeId);
$storeManager->setCurrentStore($store->getCode());

In this case $store->getWebsiteId() should be actual (selected) website id. But this can be done only in case you use the store block which works in the same way as a standard magento block (to select a store scope):

Here is my result in the example when I select the default store view (id == 1):

and here is for the default global scope (All Store Views, website id is 0):

Update:

You can add a method like this in the helper and use it where you want:

/**
 * @var \Magento\Framework\App\State
 */
protected $state;

/**
 * @var \Magento\Store\Model\StoreManagerInterface
 */
protected $storeManager;

/**
 * Data constructor.
 * @param Context $context
 * @param \Magento\Framework\App\State $state
 * @param \Magento\Store\Model\StoreManagerInterface $storeManager
 */
public function __construct(
    Context $context,
    \Magento\Framework\App\State $state,
    \Magento\Store\Model\StoreManagerInterface $storeManager
) {
    parent::__construct($context);
    $this->state = $state;
    $this->storeManager = $storeManager;
}

/**
 * @return int
 */
public function resolveCurrentWebsiteId()
{
    if ($this->state->getAreaCode() == \Magento\Framework\App\Area::AREA_ADMINHTML) {
        // in admin area
        /** @var \Magento\Framework\App\RequestInterface $request */
        $request = $this->_request;
        $storeId = (int) $request->getParam('store', 0);
    } else {
        // frontend area
        $storeId = true; // get current store from the store resolver
    }

    $store = $this->storeManager->getStore($storeId);
    $websiteId = $store->getWebsiteId();

    return $websiteId;
}

Result should be like this:

1. Backend without scope selector - 0

   

2. Backed with scope selector - selected website id or 0 if nothing selected (All Store Views)

   

3. Frontend - current website id

   

If you are looking for the short-way on the frontend:

You can use the Magento\Store\Model\StoreResolver class for that purpose:

/**
 * @var \Magento\Store\Model\StoreResolver
 */
private $storeResolver;

/**
 * @param \Magento\Store\Model\StoreResolver $storeResolver
 */
public function __construct(
    \Magento\Store\Model\StoreResolver $storeResolver
) {
    $this->storeResolver = $storeResolver;
}

/**
 * Returns the current store id, if it can be detected or default store id
 * 
 * @return int|string
 */
public function getCurrentStoreId()
{
    return $this->storeResolver->getCurrentStoreId();
}

It is correctly detects the currently selected store view on the frontend area, but useless for the admin area.

These error comes from a method public function getFormattedAddress(Address $address) because you expect to get the order address as a first argument. Try to remove the argument validation:

public function getFormattedAddress($address) {
    return $this->addressRenderer->format($address, 'html');
}

However if you are using the \Magento\Sales\Model\Order\Address\Renderer class there should be Order Address, so I'll recommend you to use the Magento\Quote\Model\Quote\Address\ToOrderAddress model and convert your quote address to the order address:

public function getFormattedAddress($address) {
    if ($address instanceof \Magento\Quote\Model\Quote\Address) {
        $address = $this->quoteToOrderAddressConverter->convert($address);
    }

    if (!$address instanceof \Magento\Sales\Model\Order\Address) {
        throw new \Exception(__('Expected instance of \Magento\Sales\Model\Order\Address, got ' . get_class($address)));
    }

    return $this->addressRenderer->format($address, 'html');
}

where the $this->quoteToOrderAddressConverter instance of Magento\Quote\Model\Quote\Address\ToOrderAddress. Add it in using dependency injection (prefered) or using the ObjectManager.