Actually, there is a feature in core OpenGL that will give you detailed debug information. But you are going to have to set your minimum version requirement pretty high to have this as a core feature.
Nevertheless, see this article -- even though it only went core in OpenGL 4.3, it existed in extension form for quite some time and it does not require any special hardware feature. So for the most part all you really need is a recent driver from NV or AMD.
I have an example of how to use this extension in an answer I wrote a while back, complete with a few utility functions to make the output easier to read. It is written in C, so I do not know how helpful it will be, but you might find something useful.
Here is the sort of output you can expect from this extension (AMD Catalyst):
OpenGL Error:
=============
Object ID: 102
Severity: Medium
Type: Performance
Source: API
Message: glDrawElements uses element index type 'GL_UNSIGNED_BYTE' that is not
optimal for the current hardware configuration; consider using
'GL_UNSIGNED_SHORT' instead.
Not only will it give you error information, but it will even give you things like performance warnings for doing something silly like using 8-bit vertex indices (which desktop GPUs do not like).
To answer another one of your questions, if you set the debug output to synchronous and install a breakpoint in your debug callback you can easily make any debugger break on an OpenGL error. If you examine the callstack you should be able to quickly identify exactly what API call generated most errors.