Yes, AJAX Shopping Cart allows for quantity update right in a pop-up window. For this, you need to set 'Allow to change product quantity in confirmation dialog' to yes in the AJAX cart general settings. Read on to see a screenshot.
Yes, once a product is added to cart, a client can either go to the shopping cart or continue shopping. If they choose the latter option, they can stay on a product page or go to a category page. See the complete instruction to make all the settings correctly.