I have always seen (and used) "lay" as the verb of choice.
My own preference would be to indicate which direction "flat"
is; i.e., to make it clear which side sits on the flat surface
for unpacking.  Flat, like everything else, is a relative term!

It also strikes me that to "lie" flat for unpacking, sounds more
like an instruction for the unpacker, than for the crate.  I have
seen some bizarre packing in my time, but so far I have not yet
had to lie flat to unpack anything.

In the same vein (but different subject), since most of the people
I hire I have to work with directly, I look for a compatable sense
of humor (or appreciation of the absurd) and ability to work in
the chaos that a busy registrars office engenders.  Finding a job
in the museum world is just like finding a job anywhere else: right
skills and experience *and* right place at the right time.

Janice Klein, The Field Museum
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