When the option to use randomly generated names is checked in the /z/ manage thing, nothing shows up in the name field. Why is this?

It's a rule generator thing. The base rule is *, and you can expand any other rules with the percent sign.

* => %F %L
F => Alice|Bob
L => Smith|Jones

I should probably put something like this onto that page...

Okay, cool. Next question: how would I do something like, say, last names generated from multiple parts?

Same way.

* => %F %L
F => Alice|Bob
L => %A%B
A => some|things|that|form|the|first|half
B => stuff|forming|the|second|half

Then you'd get names like "Alice halfstuff" and whatever.