it'd be nice to e.g. figure out if firefox is running:

  pidof firefox

However...how to do this from X when you don't have a terminal handy?

You could use e.g. synapse...but then the shell just closes.  you could do
in synapse `xmessage $(pidof firefox)` but that's awful.  Instead
either a synapse handler or something should be written to solve this
problem.