The Frame stores images in the cloud. Were the internet to go down, whatever photos that are on your frame would continue to display (and also remember they're still safely stored in the cloud). The only difference is that, so long as there is no internet connection, there would be no new updates going to your frame. But that wouldn't affect the current slideshow on the frame, that simply continues to play.
The nixplay software directly accesses your Picasa account. However, by being logged in to Picasa, you then need to tell the Picasa software that you want to sync it with nixplay. Once you have signed in to Picasa, you will have the option to sync specific folders to the web, which will show up in the nixplay website.
Not yet. After uploading the photos you can sort them into different slideshows which are then added to the frame through nixplay's website. As of now the photos show in the order they are put in the slideshow. Nixplay says they are working on a function to play the pictures multiple different ways including by date or alphabetically, but that is not a function quite yet.