I What would a universe look like if it had 2 time dimensions?

If there were two time dimensions, and our experience only recognizes only one time dimension, all objects' time trajectories would either be completely overlapping or completely non-intersecting with one another. Otherwise, we would see objects popping in and out of existence when their time trajectory overlapped with our own.
One might imagine this as a sci-fi pretext for parallel universes, but really, this is speculation in the extreme.
No physics I am aware of requires two time dimensions.
Though it would be possible from a purely mathematical standpoint to do relativity with two time coordinates. I don't think anything sensible or worthwhile comes from it.

 

I think you already know the answer. Causality requires only one time dimension. How the universe "would" look like if there were n spatial dimensions is incorrect. You mean how the universe looks like with n dimensions. The universe already has more than 4 dimensions according to string theory. Not because it is an interesting question to imagine a n-dimensional world, but because the dimension of spacetime falls out of a calculation. It's not a free parameter. Either 8 or 24 for massless fields, that is 10 or 26 for fermions. This is what you get from requiring Lorentz invariance of the string.

 

One could probably write a science fiction book on the subject, or maybe a series.

I don't think we completely understand the arrow of time, but let's just say it is locally pointed in the direction of increasing entropy. In our universe, this is in the direction of increasing time. The arrow of time doesn't necessarily have to point in the direction of time, but for whatever reason, our cosmology is such that everything is homogeneous and isotropic in the space directions, but very different along the time direction. Maybe there's a reason, but I don't know. If there are two time dimensions, entropy might increase in one time direction but be constant along the other. In which case, time would flow like it does in our universe, but the geometry of space-time would be different. Or entropy might increase in both directions like the radius of a circle: ##S = f(\sqrt{t_1^2 + t_2^2})##. In which case, time would appear to flow "outward" from the center of time, which would be a special point like the Big Bang.

If the arrow of time is given by gradient of entropy, then there are no closed arrowtime-like curves. Closed coordinatetime-curves aren't really a big deal for causality.