I can think of so many issues here I hardly know where to start.
You can't supply a clock signal to the Z80 from an IO port, you need a high frequency, about 2-4MHz.
The Z80 has NO IO pins, everything is memory or IO mapped at high speed. There is no practical way to directly connect the two chips. You will need latching hardware but the Z80 still needs ROM and RAM to run.
But the Z80A started to get very hot
The Z80 may just run hot, it's a very old chip.
I've had hard times to find it, I'd rather not burn it!
Z80s are still produced and easily found.
when I spy the addresses required by the Z80A,
What do you mean "spy the addresses"?
I've just written a program for the arduino to send 0xC3 (JP) when the 0 address is requested, or 0 for any other address, so it will code a JP 0 and forever loop.
How will you do this? You have to test the 16-bit address against 0000, turn the data direction around on an Arduino port, then write 0xC3 followed by 0x00 and another 0x00. All within the cylce time of the Z80.
If you just want to test the Z80 you can easily do that by pulling all the data lines constantly low (00 = NOP), the Z80 will cycle the entire memory map and you can view the address lines with a scope.
I must say I love the idea of building a Z80 computer, just for the shear retroness of it, but I don't see how an Arduino can really help unless you want to make what is affectively an EPROM emulator.
In this case the Arduino can be very useful.
To do this your Z80 needs RAM and IO chips and per normal, it also needs an EPROM to run the code from but you substitute a RAM for that and dual-port it with some octal buffers. Now you have an EPROM emulator and you can compile Z80 code and download the HEX file to your Arduino which in turn dumps it into the RAM that is emulating the Z80s EPROM.
It's much easier than I probably made it sound.
I've done similar many times (a while back I admit) and can help if you really want to do it.