For investigating performance issues, you need some kind of profiling data. You could build some simple profiling into the game, or you could ask people to run an external tool. I'd also suggest asking users with issues to run dxdiag and send you the log file - that will tell you their system specs in some detail, including driver versions.

If it's only a small number of users with problems then it could be that they have hardware problems of some kind. For example: https://randomascii.wordpress.com/2013/08/06/defective-heat-sinks-causing-garbage-gaming/

For crashes, what you want are crash dumps from users computers. One option to obtain those dumps is to ask them to enable http://msdn.microsoft.com/en-us/library/windows/desktop/bb787181(v=vs.85).aspx temporarily and send you the dump files. You should then be able to load them in a debugger and work out what's gone wrong. For them to be useful you need to have enabled pdb file generation when you compiled the game (but you don't need to distribute the PDB file). Also see https://randomascii.wordpress.com/2012/07/22/more-adventures-in-failing-to-crash-properly/ if your game is exiting unexpectedly without generating a crash dump.

I didn't read the entire thread in detail, but the key question here is - can't you enable logging on your code? It's kinda like running untested code on your own system - if you encounter a problem, you debug it in one way or another. Having different debug levels in your code will, in the very least, give you a general idea where something is failing.

For the the cases where performance tanks on a similar system, can you do remote profiling? Eg create a demo scene and record timings for it manually and then compare them to some ground truth you would expect a system of that magnitude to generate?