Sorry if I missed something, but what happens when you insert the dead test cart?
(I would recommend the new destest as it can actually display meaningful text even if all RAM is broken, it displays it from ROM and uses hardware tricks to make the VIC chip display/hide parts of the text to indicate fail/pass and whatnot for the initial ram test, instead of just flashing error codes like the old dead test does).
A bit of theory of operation:
The Z80 starts up first. It has to, there is a hardware bug in the Z80 that makes it do random memory accesses if it isn't allowed to do it's normal startup procedure!
The Z80 uses 4k of the C128 Kernal ROM, the other 12k is used by the 8502 in C128 mode. The Z80 code first checks if the GAME and/or EXROM signals are held low in the cartridge port, and I think it also tests if the Commodore key is held down. If so it switches directly to C64 mode. Otherwise it does some more initialization and then switches to regular C128 mode, and I think the C128 8502 Kernal checks GAME and EXROM again once or twice and uses that to switch to C64 mode.
Note that there is a signal telling if you are in C64 or C128 mode. That signal should obviously start in C128 mode, and if you have a cart installed and/or hold the Commodore key down it should switch to C64 mode almost immediately after power on. If this works then you can be sure that the Z80 executes code with a decent success rate.
A few things that you can easily test:
With a Datasette connected, or a test setup for the sense wire, the motor power should come on a brief moment when powering up and then be switched off. If you pull sense low the motor signal should come on. This indicates that the Kernal 60Hz interrupt code and the I/O ports work correctly.
If you then type LOAD and press return, or press shift+RUN/STOP while having sense active, the motor should still be on. Pressing just STOP (RUN/STOP without shift) should then turn off the motor signal. This tells that the Basic interprester correctly calls the Kernal cassette LOAD code.
Also if you connect any disk drive set to the default device 8 the C128 reads the first sector on the first track, which is the way it boots for example CP/M. If this happens the C128 Kernal runs.
Re garbage: I don't know if there are any dead test style diagnostic ROMs that can use the 80 col output. You might want to check if the sync frequencies are correct after starting without a C64 cart inserted, that would indicate that the Kernal at least ran the code that initializes the 80 col chip. The 80 col chip has it's own RAM and also it's own clock, so if you get the correct sync frequencies then it's likely that any diagnose thing for the C128 would correctly display things on the 80 col output.
I've toyed with the idea of creating a C128 Z80 diag rom of sorts, like a cross between the stock C128 Z80 ROM code and the modern RAM tests that for example Adrians Digital Basement have used to diagnose and repair various TRS 80 computers in his videos. I don't know if there would be a real use case for a test like this as compared to a C64 dead test / destest cart, but still. In general the dagnose+repair procedure seems to be to get a C64 dead test / destest to work correctly, and then move on to whatever other symptoms may exist.
Re Z80 clock frequency: I can't remember how the circuit works but the C128 slows down the Z80 to fit 6502/6800 style bus cycles, and maybe it does this by just elongating the Z80 clock pulses rather than using the Z80 inputs that can extent bus cycles (wait state input)?
As for the AC voltages, it's normal that they are higher than spec depending on the load of the PSU and whatever your mains voltage happens to be at the moment.
Side track: The stock C128 power supply is a good power supply, so if you eventually want to use this C128 without needing to share PSU with your C64 you might want to consider getting a stock C128 PSU. You might want to replace the capacitors though, IIRC there are only a few of them and they tend to be from one of the bad brands (IIRC FRAKO or ROE or similar, the yellow/orange/red european ones tend to be bad, while the blue Philips ones (obviously found in for example the Philips made versions of the 1084 monitor and whatnot) tend to be good).
Fun little known fact: If you lower the 5V input from the external PSU the jail bars on the 40 col output goes away almost completely. This is a weird reason for not using contact cleaner on a working C128, by wiggling the power connector you can get rid of the jail bars, while if you use contact cleaners you will have jail bars forever
