Our application is attempting to write a file to a UNC folder using an ASP.NET web service running under .NET 4.5, IIS 7.5, and Windows Server 2008 R2. Howeve
I had similar problem accessing a network share using AppPoolIdentity in an ASP.NET application (access denied). Using NetworkService account or other domain account worked but these were not the best solution. I performed almost all the tests you did but finally found something that worked.
I figured out that the Network Service account was not used when accessing the shares, just like you did (i expected domain\machine$ account)
This worked for us: On your IIS web site, go to Authentication and change the Anonymous Authentication item to "Application Pool Identity". It's by default set to "IUSR". This solved our problem.
Also maybe activating ASP.NET impersonation (still in Authentication menu) may help.
Thibault