It is clear that you have a problem with your internet connection.
But the weather gadget shows something I have experienced before for no apparent reason, so let's test something:
Close all the gadgets on the desktop and make sure the sidebar.exe process is closed in the task Manager. Then go to C:\Program Files\Windows Sidebar and take ownership of wlsrvc.dll
Back up the .dll somewhere just in case, then download the attachment
wlsrvc.rar I provided and extract it to the Windows Sidebar folder, overwriting the original .dll
Then try and start the Weather gadget again.
One more thing I need to ask? Is this a new install of Windows 7? Did you upgrade from Vista or did a Clean install of 7?