There isn't really a way to implement carts and checkout outside of Shopify. At least there isn't if you want to leverage Shopify's checkout infrastructure. You *could* take a look at the Order creation API, though keep in mind that doesn't offer any of the protections that a merchant would normally get when an order is placed through their checkout.
Apps aren't hosted on the App store, they are simply a distribution mechanism for application developers. If you don't have any specific need to be on the app store you can use a private app instead.