I tried to find a built-in for geometric mean but couldn't.
(Obviously a built-in isn't going to save me any time while working in the shell, nor do I suspect there's any difference in accuracy; for scripts I try to use built-ins as often as possible, where the (cumulative) performance gain is often noticeable.
In case there isn't one (which I doubt is the case) here's mine.
gm_mean = function(a){prod(a)^(1/length(a))}
No, but there are a few people who have written one, such as here.
Another possibility is to use this:
exp(mean(log(x)))