Thoughts & questions on custom CPU architecture
Is it overkill to have every register inputting to the ALU, or is it
  more sensible to have one input hardwired to, say, A?


You will need more 'buffers' and connecting logic to route all the registers to the ALU, so you its a space vs functionality trade off. 


  This design is as you can probably tell influenced by the Z80 (hence
  the register naming). Is it a good idea to base the CPU design off of
  such an old CPU or does it not make any difference really?


Older CPUs are easier to understand because they have reduced instruction sets and smaller data widths, I would prefer to design with an old CPU if I were implementing this myself. 


  I've not decided whether to have a simple instruction register or
  whether to implement a pipeline. What kind of performance gains might
  I be able to achieve in a simple CPU design like this? i.e. is it
  worth the extra effort of constructing a pipeline?


If you don't want to wait for your ALU to complete instructions, you'll want a pipeline.