By "optimal expansion speed", do you mean "maximum possible expansion speed given particle physics, general relativity and thermodynamics (according to our current understanding thereof)", or do you see some reason that a slower expansion would be beneficial to the ultimate goal (or is that the question)?

Avoiding heat death may be beneficial, for example. As I wrote to Mitchell Porter, to me, the most interesting special case of the question is: if you want to build the fastestest computer in the Universe, should it expand with the speed of light? I'm really not a physicist, so I don't even know the answer to a very simple version of this question, a version that any particle physicist should be able to answer: Is it possible for some nontrivial information processing system to spread with exactly the speed of light? If not, what about expansion speed converging to c?

what properties are necessary for a structure to seem like a universe to you in the first place?

You (and Mitchell Porter) are completely right . At this point, I don't have a convincing answer to your obvious question. In the meantime, Tegmark level IV is a good enough answer for me. (Note to Mitchell: it would be very hard to find someone less platonist than me. And I find Tegmark's focus on computability totally misdirected, so in this sense I am not an intuitionist either.)

I'd be pretty flexible with it, but I think I'd require it to have something timelike, some way for conditions to dynamically evolve over at least one dimension.

I think we disagree here. Please see my answer to Mitchell about the emergence of time from the more basic concept of memory.