Is the extension compatible with Checkout by Amazon or AmazonPayments (both Creativestyle) ?

No for AmazonPayments.

Yes for Checkout by Amazon. Please use the below workaround when installing Fooman Surcharge with Checkout By Amazon:

1. Edit the following file: /app/code/community/Creativestyle/CheckoutByAmazon/controllers/CheckoutController.php

2. Look for the following code: 

 foreach ($order->getItemsCollection() as $item) {
     $productId = $this->_getProductId($item);
     if (!$productId) continue;
          $product = Mage::getModel('catalog/product')->load($productId);
          $amazonItem = Mage::getModel('checkoutbyamazon/api_model_checkout_purchaseItem', array(
              'MerchantItemId' => $item->getId(),
              'SKU' => Mage::helper('checkoutbyamazon')->sanitizeSku($item->getSku()),
              'Title' => $item->getName(),
              'UnitPrice' => array('Amount' => Mage::helper('checkoutbyamazon')-
>sanitizePrice($item->getBasePriceInclTax())),
              'Quantity' => round($item->getQtyOrdered())
           ));
     $amazonItem->setItemTax($this->_getAmazonPrice($item->getBaseTaxAmount()));
     $itemList->addItem($amazonItem);
     unset($product);
 }

3. Directly after the code above, add the following code: 

 if ($order->getBaseFoomanSurchargeAmount() != 0) {
     $amazonItem = Mage::getModel('checkoutbyamazon/api_model_checkout_purchaseItem', array(
         'MerchantItemId' => 'surcharge',
          'SKU' => Mage::helper('checkoutbyamazon')->sanitizeSku($order->getFoomanSurchargeDescription()),
          'Title' => $order->getFoomanSurchargeDescription(),
          'UnitPrice' => array(
             'Amount' => Mage::helper('checkoutbyamazon')->sanitizePrice($order>getBaseFoomanSurchargeAmount()+$order->getBaseFoomanSurchargeTaxAmount())),
              'Quantity' => 1
          )
    );
     $amazonItem->setItemTax($this->_getAmazonPrice($order->getBaseFoomanSurchargeTaxAmount()));
     $itemList->addItem($amazonItem);
 }