SharePoint and Office Integration using different browsers?

Well, the short answer is that its not supported, in general. It has nothing to do with APM, and everything to do with supported browers. A large majority of SharePoint components require ActiveX, its also important to note that x64 versions of IE will have problems as well. Are you attempting to open documents directly, or via Office Web Apps? Which version of SharePoint? Office Web Apps supports more browsers, because it doesnt directly require office integration activex controls. http://office.microsoft.com/en-us/support/supported-browsers-in-office-2013-and-office-365-HA102789344.aspx

There are workarounds for NON-IE browsers, you can find more details at The following link: http://technet.microsoft.com/en-us/library/cc263526(v=office.14).aspx

You could also try the NPAPI plugin available here: https://code.google.com/p/npapi-msdocs/