Why is cyclooctateraene shaped like a "tub" rather than a "chair"?
It's all about the 3D structure of double bonds.

If we look at the tub form we see that all dihedral C-C=C-C angles are 0°. the C=C-C angles are 125°, also pretty close to the optimal value. Everything works out fine, there's basically no strain on the whole molecule.


I tried to build the chair form of all cis, but every optimization ended up in either the all cis tube form or the cis-trans-cis-trans chair form. The best structure that looks like the all cis chair form I ended up with is this:



First of all, it's pretty flat, so antiaromaticity would again play a role and the dihedral angles are also ~33° for the double bonds in front and in the back, which is energetically quite bad and unstable.

The cis-trans-cis-trans compound is definitely no monstrosity. I'm working a lot with trans-cyclooctenes and I don't see why this shouldn't be possible if a trans,trans-cycloocta-1,5-diene is possible. 



Here we've got 0° for the dihedral angles for the cis bonds and 125° for the trans. Yes, this is far away from the optimal value of 180° but quite similar to what you expect from trans-cyclooctenes (around 135°). I feel like it would be quite reactive and unstable but it could definitely exist. There's just no good way to produce it I guess.

So overall: all angles in the tub form are much closer to the optimal values than in either chair form and the all cis form is quite planar.