OK, I rebuilt Qt 5.5.0 after removing the environment variables, but it unfortunately did not solve the problems I described.
I was ultimately able to get the VS addin 1.2.4 working in VS 2013 by specifying the Qt installation as being at C:\Qt\5.5.0\qtbase instead of just C:\Qt\5.5.0. I was able to create a simple Qt GUI program and run both Debug and Release builds.
But the remaining problem is that all Qt tools like designer, assistant, and linguist all crash as soon as they are started. In each case the event log holds an entry like this,
Faulting application name: designer.exe, version: 188.8.131.52, time stamp: 0x55d4a1df
Faulting module name: Qt5Gui.dll, version: 184.108.40.206, time stamp: 0x55d48d34
Exception code: 0xc0000005
Fault offset: 0x00000000006850f5
Faulting process id: 0x5b8
Faulting application start time: 0x01d0daadb7408ac1
Faulting application path: C:\Qt\5.5.0\qtbase\bin\designer.exe
Faulting module path: C:\Qt\5.5.0\qtbase\bin\Qt5Gui.dll
Report Id: f504ee37-46a0-11e5-8267-8cdcd42c37a6
Faulting package full name:
Faulting package-relative application ID:
So there is still some kind of issue with my build, though my simple Qt GUI project was evidently able to use Qt5Gui.dll without a problem.
Any idea what could be causing this? Is there a problem with the configure options I chose? Or with the modifications I had to make to qcompilerdetection.h to get the build to work?