As previously stated, the HC00 should not be present...
The interesting thing (from looking at the used gates of U4 - 74LS32) is that only 2 of the 4 gates are used. The unused gates do not have their inputs connected to anything (either 0V or pulled up to +5V). This can introduce noise and strange behaviour into the used gates on the package - and this IC drives the address decoder (U6) that selects the ROM, RAM, and keyboard.
This is poor (original) design - and a faithful copy of the original will contain the same fault. I would be tempted (myself) to connect the unused input pins of U4 to 0V.
If the CPU cannot correctly read the instructions out of the ROM, it will go loopy and can halt.
The same address decoder (U6) is used for everything.
So, if the address decoder fails (or there is noise or timing issues) then all sorts of random things will go wrong (i.e. the computer will be unstable).
This is where I was heading in looking for the timing error(s) that are causing misreads of the keyboard (and hence the ghost writing).
This can be either U6 or U4 to blame.
However, in order to check the signal timing, we have to have a Z80 CPU that runs (the chicken and the egg scenario).
If the Z80 has stopped running, it is pointless to complain that you can't measure the /MREQ signal on pin 19 of the Z80 CPU.
You will have to power down the machine, count 10 seconds, and then power it back up again to (hopefully) reset the Z80 CPU and kick it into running.
If you get the READY prompt and the ghost writing then you can use your oscilloscope. If you get a picture of a black cat in a coal cellar at midnight, the Z80 is not running. This design requires an operational Z80 CPU to generate the video characters.
Remember the trick with the PET and the 6502 CPU? We always check pin 7 (SYNC) for signs of activity. If none are present, the CPU has halted.
Likewise with the Z80 CPU, if /MREQ is not pulsing away - the CPU is not fetching and executing instructions, so your machine is a door stop and there is no point in poking around with an oscilloscope.
Dave