RE: Upgrading GP10 to GP2013 Webaccess
I'd recommend to upgrade to GP2015 as web client is much better in v2015 than v2013. With GP2015, web client supports any OS (I have personally used it on Windows and Mac) as long as the OS supports Silverlight.
However, it may not work well with Windows XP. It may not work well on Mac OS. Underlying technology is Silverlight, so you may face some issues with Silverlight plugin.
You may also want to analyze the customizations that you are currently running and see if they can be upgraded to be supported on web client. Especially, if you are using Crystal Reports, you may have to migrate them to SSRS.
Apart from simple issues, GP2015 Web Client is quite stable and can be embraced for production usage.