Way to have Dropbox remember the last page I was on in a PDF file

You're going to have to relax atleast couple of those restrictions. Security + Sharing + Multiple devices + Easy - I doubt any service exists which meets all of your criteria.

Having said that, SpiderOak is probably the closest that comes to meeting your requirements.

There is no mention of the supported version number on either the Google Docs Viewer tool page or the View files in Google Docs page from Docs Help.

I don't know if it is possible to give a complete answer to your question but my best guess is that there is not any officially supported version, but rather that the team tries to keep the tool as up-to-date as possible and to support as many features of the latest version of PDF as possible.

Since the viewer is provided as a web service, Google doesn't necessarily have to think about supported versions in the traditional way. They may be closely monitoring usage and as they receive error reports during rendering of documents from the web service, they may be working to resolve those problems so that the percentage of successful views is as high as possible. After all, in my experience there have always been vendors who "supported" a particular version of PDF but which did not successfully render particular documents due to bugs in the rendering engine - but the existence of bugs didn't change the officially supported version number.