How do I remove download links/button redirect to accurate download

Question

When I go to some sites that have a download link underscored in blue or a button next to text or any other part of the page, I was able to click and download the file. When you point at the link or button you would see the path it will take to download the file. It would have the website name your visiting plus other codes.
But on returning to these sites, when I choose what I want to download, the link or button wants to go to accuratedownload.com. It presents the name of the item I want and with a .EXE extension. My main browser is Firefox version 26.0 on my XP PC. I test Internet Explorer and it produces the same results. Something has changed. I go to the site on my mobile phone, I am able to download the file from the same site and the links/buttons are not embedded with codes to go to accuratedownload.com. I do not know what to call this problem since I had no luck in finding solutions to redirection being embedded in a link or button on a web page that worked fine.
I've tried a reset Firefox, my profile, running Symantec Anti-Virus, Malwarebytes Anit-Malware, Clearing history and so on. The problem still occurs. Need Help.
Note: It would be nice if when the Download Screen pops-up, that we could rename the download file to a name we want (Saves time). Even if you rename the file, the history should record where it came from with the original name if it needs to be downloaded again.

Answer 1

To let you know, When I did a reset and restarted Fire Fox without the Add on's. I tested each user account on my PC. Each user account that used Fire Fox or Windows Explorer showed the same symtoms.
Back to troubleshooting, I've completed a scan using Sothos Free Virus Removal Tool. After a lengthy scan of every drive, partition, USB Thumb Drive and so on that it found, in about 5.5 hours it found Mal/FakeAvCn-C, Mal/JavDlEx-A and CXmal/BadLnk-A and continued by clicking the Clean Up button and that took about 15 minutes for some reason. Then it asked to send the log to Sophos. I clicked send and it failed to get to them.
I tried going on the internet to test the cleanup by clicking on some links and buttons. I could not get on the internet. I found that the NIC card was not working. Don't know why now but probably was affected by the scan. I found myself troubleshooting and reinstalling NIC drivers several times before the NIC card started to run correctly.
I read you post and tried to change the proxy settings in Fire Fox to "No Proxy." The scan, clean up and proxy settings change did not remove the embedded paths in links or buttons that allow you to download or be sent to the intended location. It still had paths to accuratedownload.com and eachdownload.com.
Question: By default is the proxy setting suppose to be "Use system proxy settings"? Should I put this back to this setting.
Windows Explorer has the same problem. Is Fire Fox a stand alone product or does it build parts of itself using Windows Explorer?
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Besides the scans and removal done to the registry, the scans it did on the files and folders, the Sophos log shows the virus attached to files in Documents and Folders, All Users (and other user accounts), Application and data folders, Java and Cache, including any shortcuts and long cryptic files like (pg42l784fcuo452hsb6cfbkoke045p75f2e02887m).
When programs fail to delete or cleanup after themselves, what locations can be safely deleted manually?