Hello Bruno,
Version 1.x of the SDK does not support customers. Version 2.0, on the develop branch, does. We will release to master soon, and release updated docs at the same time. Version 2 is a significant change, so please review the code before jumping in.
If you want to persist the user's cart, you will have to do that yourself. You can convert the cart into JSON to save to disk. On app resume, create a new cart and initialize it with the JSON.
If you create a checkout token, you may wish to re-use that, to avoid creating abandoned checkouts. The token is created by Shopify when you POST the checkout the first time. You may want to keep the cart and checkout in sync. The easiest way is to recreate the checkout from the cart, and then set the token on it, and then update the checkout on Shopify.
It is up to your app to decide when to re-use a checkout, and when to discard it and create a new one. Usually when the user empties the cart completely, you would probably want to delete your locally saved cart and checkout token.