@Roffen or anyone
You should not have any issues downloading the Firefox installers as long as you have XP SP3. The certificate errors would be due to outdated XP service packs, not outdated Internet Explorer.
I have downloaded Firefox on dozens of machines with Internet Explorer 6, 7, and 8 with SP3. This is successful because SP3 contains updated CRYPT32, RSAENH and other cryptographic files and libraries which are required for SHA2 support, among many other improvements and fixes.
For example, if you attempt to open secure websites such as Outlook Webmail on IE 6 with XP SP2, it will fail. But, with SP3 it will succeed. (You should also change IE's settings to TLS instead of SSL because of the POODLE vulnerability).
http://sdfox7.com/xp/files/ol365ssl.jpg
http://sdfox7.com/xp/files/ol365tls.jpg
For future reference, you can download XP SP3 directly from my FTP or the Internet Archive: http://sdfox7.com/xp/sp3/windowsxp-kb936929-sp3-x86-enu.exe or https://web.archive.org/web/20140119014510/http://download.microsoft.com/download/d/3/0/d30e32d8-418a-469d-b600-f32ce3edf42d/WindowsXP-KB936929-SP3-x86-ENU.exe
I also have the installers for Internet Explorer here (IE 6 is included in XP so the download link is unnecessary):
Internet Explorer 7: http://sdfox7.com/xp/sp2/IE7-WindowsXP-x86-enu.exe or https://web.archive.org/web/20130509183828/http://download.microsoft.com/download/3/8/8/38889dc1-848c-4bf2-8335-86c573ad86d9/IE7-WindowsXP-x86-enu.exe
Internet Explorer 8: http://sdfox7.com/xp/sp3/IE8-WindowsXP-x86-enu.exe or https://web.archive.org/web/20130510043925/http://download.microsoft.com/download/C/C/0/CC0BD555-33DD-411E-936B-73AC6F95AE11/IE8-WindowsXP-x86-ENU.exe