Welcome to the forum, ChrisG!
Starting from the second question:
According to
this datasheet, page 80 or 82, the maximum clock pulse width, low, is 2000ns, at 50% duty cycle that would make 250kHz (ignoring rise and fall times), so you are already lucky to have got it down to 10kHz.
You might try changing the duty cycle while lowering the frequency.
If you have a CMOS version of the Z80, though, there should be no such limitation.
[...]while ground was common, had a lower ground signal then 0v
If the ground was common, you are probably referring to the lower part of the output square wave.
In general, the output of a signal generator is centered around 0V, and that's the reason you had to add a 2V offset: to bring a +2/-2V output (4Vpp, ground centered) to 0/+4V.
If the connection was direct and the signal generator has low output impedance, it's possible you fried the Z80 by exceeding the maximum current on the input clamp diodes, reading the SG manual it seems that the HiZ setting is only useful to give correct output voltage reading, and is not changing the actual output impedance.
The Z80 datasheet at page 79 states an absolute minimum of -0.3V at any pin, while you were applying -2V.
So, a raw estimation of the current (assuming 50ohm SG output impedance): (2-0.3)V/50ohm = 34mA, quite enough to damage the chip.