What other purposes can I use my [obsolete] hashing hardware for? It's pretty easy to reuse CPUs or Video cards that were previously used for Bitcoin hashing, but what can I do with FPGA hardware when the difficulty becomes too high for it to be worthwhile to mine?

Are there competing networks I can mine for?  Are there other technologies (like email/hashcash) that can reuse the hardware with little modification?

Is there a similar way to recycle old ASIC's when the time comes?
FPGAs and GPUs are reprogrammable, so you can use them for anything. FPGAs require some specialised programming, while GPUs just use OpenCL.

ASICs have specialised chips highly optimised for mining and only mining. I don't think they could ever be reprogrammed or reused to do anything else than mining blocks according to Bitcoin rules.

As for what to do with that hardware, have you considered such noble causes as Folding@home, or perhaps less noble split-key vanity mining, for example through my Vanity Pool website?