Well, at this point, I don't really see much going on in your log. I can't quite explain your Google redirects, but the fact that you couldn't get online with Safe Mode With Networking makes me think it might not be a software problem. And when an infection redirects your browser, it usually goes to a spam site, not Google. How does this particular computer connect to the internet? You said you have a wireless network; does this computer connect wirelessly as well? Or does it use a cable to connect directly to the router (or a modem hooked up to the router)?
If you connect wirelessly, do you have an adapter for the desktop or did it come equipped with a network card? If there's an adapter, try testing it with a different computer to make sure it works. If it has a network card, it may not be functioning properly and could need to be replaced or reseating (reseating involves opening the computer and ensuring that a part is properly inserted into its slot). Also, be sure to try connecting through an ethernet cable.
If you connect through a cable, make sure it is firmly inserted into the appropriate jacks. Depending on your setup, you may be able to hook up to a different jack on the router. Another thing you can try is unplugging the router for about five minutes and then plugging it back in. This doesn't sound like it will fix your problem, but you never know.
Did you recently move your computer? It's possible that it was moved out of the router's range. Do you know exactly how long it has been acting up? We could always try System Restore to roll your settings back and see if it helps at all. Also, the configuration could be incorrect, but that can be a bit more complicated, so we'll save that for later if needed.
As you can see, there are a lot of possibilities here. And there's still the chance of an infection causing significant problems. I don't see anything, but some of them can hide pretty well. Although I haven't seen evidence, you can check for a certain popular infection...
Click Start > Control Panel > System > Hardware > Device Manager > View > Show Hidden Devices.
Scroll down to “Non-plug and Play Drivers” and click the plus icon to open those drivers.
Then search for TDSSserv.sys
Let me know if you find this or not. Also let me know what other suggestions you have tried, and please answer as many questions as you can.