The crash.dmp files would be more interesting than the trace logs, were those not generated or did you remove them?

We cannot help you if you do not provide more information.

In addition to the crash dump files it would also be interesting to know how you close the dialogs, e.g. if you use the ESC key and if so, is this key also bound to stopping audio playback? Does it only happen with certain dialogs (which)?

I did not see any crash.dmp files, unfortunately. I'm trying to see if I can reproduce the crash and get some dumps.

Okay, here's the dump.
https://drive.google.com/file/d/0B1URcZnhifwbTnBETGxYRkxOQ00/view?usp=sharing

Lately, it seems that OpenMPT is choosing to hang rather than outright crash, so the audio driver keeps playing the module even though the GUI is completely unresponsive for the remainder of the running instance. Even though it is supposed to create dump files, I had to wait for the program to freeze and create the dump manually through task manager.

If OpenMPT doesn't crash, there is no reason for it to create a crash dump. A hanging application cannot know if it hangs (that would be equal to solving the halting problem), hence it cannot create a crash dump.
Anyway, the memory dump is not usable because it's a 64-bit dump. Please read on here how to create a usable memory dump:
https://wiki.openmpt.org/Manual:_Frequently_Asked_Questions#OpenMPT_crashed_but_did_not_create_a_memory_dump
Please also update to the latest OpenMPT version on buildbot.

Also, you didn't explain yet with which dialogs this happens and how you close them (mouse/keyboard).

It doesn't seem to care which dialogues in particular; song settings, usually, but sometimes it is the file browser or the global settings. I use the mouse to close them, and then a short time later the whole application hangs up. I'll be sure to read the document.

Okay, here's the 32-Bit version (I hope)
https://drive.google.com/file/d/0B1URcZnhifwbN3plVEFzZXVMalE/view?usp=sharing

I launched the 32-Bit task manager from the command line (as per the instructions), but since every computer I use is apparently haunted by gremlins, I have no way of knowing if it worked correctly for me.

I suppose it's the DAEMON Tools shell extension that shows up in the stack trace (DTShl32.dll) which is causing your issues. Try uninstalling it. I don't think there is anything we can do on our side, because there is no code in OpenMPT that would do special things that could cause a crash or hang when closing the song settings or similar.
Badly coded shell extensions are a common problem with various software, so I am rather positive that it will fix the issue. If you do not know how to get rid of the shell extension, this might help: http://daemonpro-help.com/additionally/shell_extensions/

Okay, I'll be sure to try that. Thanks.

So are the gremlins gone now or does the problem persist?

I will close this report now as the memory dump clearly showed that the DAEMON Tools shell extension was causing the hang.
PS: A no-nonsense, free and open-source alternative for mounting images is WinCDEmu, which certainly doesn't come with this kind of issues.