Is there any way to hide the vim command-line (ideally, automatically) when it's not being used? On small screens it takes up one valuable line. I'm thinking it would be 'normally' hidden, and appear when
/, etc. are typed.
No, that is not possible. The command line is always present and there's no way to hide it without changing vim's sources.
It's not only used if you type
/ but also for status messages (try
ctrl-g, for instance).