How do I troubleshoot sudden random shutdowns?

Ok, now as I suggested, make a note of the time your PC black screens, then use that event viewer to see if you get event id 42 then, and if not, see what significant event id's occur around that time.

You can research event id's here:
http://www.eventid.net/

However, as you can wake your PC easily when it DOES sleep, then it would seem your black screen isn't sleep... hence the need to see what does happen then.

(All this on the understanding we have eliminated thermal shut down, which is sudden).

Hi, you're getting a huge number of serious disk related errors. I suggest dealing with those as a priority.
Note: Hard Disk Sentinel appeared to report disks as good- that's the physical disk. The errors refer to the integrity of the NTFS file system- a different issue.

(Please confirm HD Sentinel's GUI showed all disks as good - much easier than scanning a long text file).

There's this - possibly about your memory card reader
Event ID 504 and 507 from StorDiag with Windows 10 - Microsoft Community

and this about Drive E: - which tells you what to do... same for F: & G:
Event Time : 03-Oct-18 2:52:03 PM.662
Record ID : 73843
Event ID : 305
Level : Error
Channel : Microsoft-Windows-Ntfs/Operational
Provider : Microsoft-Windows-Ntfs
Description : NTFS failed to mount the volume.

Error: {No Disk}
There is no disk in the drive.
Please insert a disk into drive %hs.
Volume GUID: {D9256266-BC77-11E5-8ABD-806E6F6E6963}
Volume Name: E:

Guidance:
The volume is recognized by NTFS but it is corrupted that NTFS could not mount it. Run CHKDSK /F to fix any errors on this volume, and then try accessing it.
Opcode : Stop (2)
Task : 6
Keywords : 0x4000000000000020
Process ID : 4
Thread ID : 2476
Computer : Trixie
User : NT AUTHORITY\SYSTEM


So you need to run chkdsk for each drive. Note the result for each and post back

E.g.
From an admin command or powershell prompt
[Windows key + X, click command prompt (admin)]
chkdsk C: /F
Your PC will need to restart.
Post back the result, which you can get after a restart as follows:
Read Chkdsk Log in Event Viewer in Windows 10 | Windows 10 Tutorials
or
How to read Event Viewer log for Chkdsk in Windows 10 [Tip] | dotTech

1) In Ccleaner uncheck:
a) Memory Dumps
b) Windows Log Files

2) Download and install: Everything

3) Use everything to search for:
a) win32k.sys-20180920-2326.dmp
b) win32k.sys-20180928-2355.dmp

Zip each and post share links into the thread using one drive, drop box, or google drive

Code:

win32k.sys-20180920-2326.dmp 21-Sep-18 7:56:59 AM     962.2
win32k.sys-20180928-2355.dmp 29-Sep-18 7:44:35 AM    904.85

4) The logs displayed a problem with a driver and that can be checked using the Windows driver verifier tool.

5) Create a brand new restore point

6) Make a backup image using Macrium: (free edition)
Macrium Software | Your Image is Everything

7) Save the backup image to another drive or to the cloud

8) Read these links on Windows driver verifier: (do not use the tool until it is communicated in the thread)
Driver Verifier-- tracking down a mis-behaving driver. - Microsoft Community
Enable and Disable Driver Verifier in Windows 10 | Windows 10 Tutorials