http://hamiltonwoodworks.com/drichards/Misc/KoaBed.jpg
Take a look at this drawing. This is for the bed I sleep on every
night. No racking or creaking and it requires no tools to assemble or
disassemble.

On to your questions:

Slats would be appropriate, especially with a king size. Our bed is a
queen and it has slats. The slats are linked together with a couple of
long webbing straps (dark gray in the drawing). The end slats have
holes drilled so they slip over dowels in the legs. The webbing ensures
the slats remain evenly spaced. It also keeps the slats together when
transporting the bed--you just roll them up and go.

I've seen other bed designs that entail pins for every slat. That seems
like a big PIA to me from both during construction as well as during
setup.

Next, yes, you should run a support from head to foot down the middle
and it should have at least one leg in the center of that support rail.
Our queen has that although I didn't show it. The center rail is shaped
like an upsidedown tee and the slats sit in the rabbet. The slats, then
are slightly less than half the width of the bed.
Hope that and the drawing give you some ideas.

Dave