Sometimes, the only rational way to isolate the problem is to start disconnecting things until the remaining circuit works properly. Once you've isolate a branch, continue with disconnecting things or bypassing individual items until you're only left with one. While a GFCI device can fail, if it turns on and trips when you hit the test button, while not an absolutely 100% check, it's close...swapping it with a new one where the same thing happens is pretty much 100%. Then, it's back to trying to isolate the problem. First thing is to carefully open and do a very thorough visual inspection of everything that's accessible. Understanding how the things works helps. You'd never want the neutral and ground tied together after the panel.