Focusing on the C# console vs Excel 2010 issue (same machine, both running as 64-bit processes), traces have been added as explained here: https://stackoverflow.com/a/25683524/403671

The initial traces from Excel and Console are exactly the same (see the question for more detail). But the last traces (posted to MSDN) exhibit this:

Console:

System.Net Information: 0 : [35268] Remote certificate: [Version] V3
[ Lots of lines describing a certificate with [Subject]=<TargetWebSite>, [Issuer]=HydrantID SSL ICA G2, etc]
System.Net Information: 0 : [35268] SecureChannel#47891719 - Remote certificate was verified as valid by the user.    

Excel:

System.Net Information: 0 : [39988] Remote certificate: [Version] V3
[ Lots of lines describing a certificate with [Subject]=<TargetWebSite>, [Issuer]=Kaspersky, etc]
System.Net Information: 0 : [39988] SecureChannel#2383799 - Remote certificate was verified as valid by the user.

It shows the certificate used in Excel is now issued by Kaspersky, which turns out to be the anti-virus product running on the PC. In fact, it's a Kaspersky product security feature to intercept communications, and this causes numerous issues, like KIS Interfering with Git, for example.

Once this anti-virus is removed everything works as expected.