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The “is not a valid Win32 application” error is caused by the incompatible operating system you are using. The Office 2013 works on Win 7 and Win 8 only. For more information, here is a site for your reference: Why can’t I install Office 2013 on Windows XP or Windows Vista?
In addition, this is a site providing further information: System requirements for Office 2013.
Since XP will no longer be supported after next spring, you should upgrade, if you look arounf some stores still have the Windows 8 upgrade for $69, it is worth it for the new speed alone.
BTW, if you have room on your drive, or install a second drive, you have the machine dual boot, so XP can stay if there are things that you can't live with-out.
Hope this helps
The web parts will work, they will work on almost anything, I used them on a Blackberry Playbook, and a Google Chromebook. But if you purchase the Premium as a cheap way to upgrade Office, then no it won't work, Office 2013 desktop clients only run on 7 an 8 as pointed out, yes you can access the web, yes you can still use Office 2010 or at least parts of it.