Is gravity time-like? Disclaimer: I'm just a physics enthusiast, so pardon my lack of full mathematical understanding.
Basically I've read quite a lot about general relativity, and I have some intuitive knowledge at least about what do we mean for spacetime to be curved, as opposed to space being curved.
I've recently come up to a video that explained that experiments to test the "shape" of space show that is virtually flat. I don't recall which video it was but the same principle is explained in the Wikipedia page for "Shape of the universe":

The actual value for critical density value is measured as 
  $ρ_\text{critical} = 9.47×10^{−27}\rm\, kg\,m^{−3}.$
   From these values, it seems that within experimental error, the universe seems to be flat.

So here is what I think about this:


*

*Does this mean that space-time curvatures are only time-like?

*Is this the cause of time dilation in regions around massive objects?

*If the curvature of space time is what defines gravity, is it just time-like? 

 A: Notational quibble: you mean "the curvature in the timelike direction".


*

*There exist observers in the spacetime modelling our Universe on very large scales who see space as flat. They still see spacetime as curved.

*That means if those observers were to "freeze time" and drag an arrow around a loop through space at any one moment, the arrow would not rotate. If they were to allow the arrow to move through time, it would rotate. Put differently, triangles drawn by those observers have interior angles adding up to 180$^\circ$, but those triangles will warp over time (specifically in the case of the flat FLRW Universe, they will expand).

*There are still other observers who see space as curved; i.e. observers accelerating with respect to the "flat-space" ones. Triangles drawn by those observers don't add up to 180 and still warp over time.

*The time dilation around massive bodies is mostly caused by the curvature in the timelike direction. However, this has nothing to do with the properties of the Universe on cosmological scales. The latter model assumes the Universe is completely homogenous, having "averaged out" over local massive bodies.

*"Gravity" is sort of a vague term, but it will be caused by the overall curvature in all directions. When fields are weak, however, the curvature in the (purely) timelike direction vastly dominates. So everyday gravitational effects like rocks falling are caused essentially entirely by this. This is why, for example, rocks fall on Earth even though triangles here still look "normal".

