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Thanks for sharing, John. The Gmail app (latest version) is indeed needed for Gmail Widgets to work.
That's right. There are solid, semi-transparent and transparent backgrounds to choose from.
As for the size issue, different launchers allocate different amounts of space to widgets. I will remove the annotations (e.g. 4x1) in one of the next releases to avoid confusion. The list and stack widgets are resizable anyway, so it shouldn't be an issue.
That's great news! About the background - is it always white even if you select another theme, like glass?
You can go into the Gmail app and make sure all labels are synched (Label settings > Sync messages > Sync: All). Also, you can wait a few hours and check for an update for Gmail Widgets (new version 3.32). 
I have just published an update. Keep in mind that it may take a few hours for it to become available for download. Let me know if it helps.
we test our apps on a variety of devices, however, we cannot cover all of them. Unfortunately, we haven't tested it on the Xperia S. If you continue to experience this issue, please send an email to requesting a refund.
you can use the refresh button (next to the refresh timestamp) to initiate a manual refresh. Alternatively, touch the counter (in version 3.31 or later). 
Can you check that you have the latest version of the Gmail app and that you have enabled auto-sync for all those accounts? You can do that from your device settings and Gmail settings.
I'm looking into it. I'll get back to you as soon as possible.