I received your screen shots. Thank you.
The graphics for each of the cloud storage services indicate you have lost the connection between your device and the 3 services: Box, Dropbox and Google Drive. The logins and passwords in question are for each of those services, not for SO2.
My first suggestion is that you use Safari to reach each of those storage services and check to be sure your logins and passwords work. If they do, you should be able to enter them in the SO2 window and re-establish your connection to each service. If they do not work in Safari, you'll have to correspond with each of the services to reset your logins and passwords.
My second suggestion is that you update your version of SO2 through the App Store. The screen shot of your original receipt doesn't indicate whether you've picked up subsequent versions. Current version for iOS is 2.3.16
My third suggestion is that since you indicate most or all of your documents are on Google Drive, install the Google Drive App from the iTunes store. If you can login from that app, you will be able to launch your documents through SmartOffice2.
Please let me know whether or not these suggestions help.
Artifex SmartOffice Support