No version of AxCrypt stores any file data anywhere else than your local device. We never upload any of your file content to our servers. On our servers we store public and (encrypted) private keys, general account subscription status and if you’re using our password manager, we’ll store your encrypted passwords there – but only if you’re using it.
The restriction on mobile apps is that we can’t encrypt or update already encrypted content on your mobile device. The files must be encrypted on a desktop device, and imported in a manner of your chosing to the mobile device. On the mobile device, they will be (temporarily) decrypted in order to be viewed.
If you sign out of AxCrypt on your mobile or desktop device, you’re always required to re-enter your password.
Not exactly sure what you mean with ‘Alphabetized’, but you can click the header and sort the list of encrypted folders alphabetically.
Since it’s free to try all features, including the mobile apps, up to 30 days the best way to answer most questions is to try it out!