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If everything else on your phone is working, and other apps install too...
Are you able to try this before someone else suggests clearing the Play Store Cache:
If you have access to a PC, log on and acess the play store here:
https://play.google.com/store/apps/details?id=com.sonymobile.themes.cover.gold
You should get an option to "Install"
If your devices is syncing, you can wait for it to Sync, then see if it installs. Otherwise you can navigate to the settings menu and go to "Accounts" and select your Google account, and force a Sync
If you get an error message, can you tell me what it says?
Thanks
Also, you don't need to plug the phone in via USB. This is a Wireless install.
You need to log in on the PC browser, and have your device selected when browsing the store
If that doesn't work, try downloading the wallpapers through the Sony Theme manager app in the display settings for the phone
Let me know if you have a problem installing these. You'll need to temporarily allow installation in: SETTINGS>Security>Unknown Sources
Turn it back on once it's done
Once you download these, put them on to the SD Card somewhere, and then use the File Manager to find the files and you can install it from the File Manager. File Commander might be bundled, if not, temporarily install ES File Explorer
Can you remove your account from the phone, and then set it up again.
Settings>Accounts>
Keep navigating in to the Account until you see the Sync options, then you can hit the Vertical Elipsses/Options Menu in the top right of the screen, and you'll see remove account
There must be a setting somewhere in the Google Accounts or a corrupt database.
Did you download those APK? or did you just try to install from the Play Store?
Are you able to take a screenshot of the error?
Upload the pictures to http://imgur.com and then post the link so we don't have to wait for the moderators