I am curious about 3-manifolds though I know little.

Here I am trying to know what invariants people in this field are interested in.

The following are what I have known and what I particularly want to know. It is also appreciated if anybody provides other information related to the question.

It seems that cohomology rings, say, with coefficients in $\mathbb{Z}$ are not quite concerned with (Am I right?). Why? I know that there is little information in the group structure. Does the ring structure give much information? I notice that the homology sphere has a trivial $H^2$ and therefore the multiplication is trivial in this case. But is this phenomenon general?

Why are people so interested in the volume (or Gromov norm) of hyperbolic 3-manifolds? Is this because the volume contains much information? I know for any given positive real number $x$, there are finitely many hyperbolic 3-manifolds with volume equal to $x$. And is this invariant well understood? By "well understood", I mean, for example, how does the volume influence the topology, how is it related to other invariants (e.g. the volume conjecture)? Also, has anything new and notable been developed in the recent work on the smallest volume of closed oriented hyperbolic 3-manifolds?

Is the fundamental group or Kleinian group still too complicated for us, given the recent work of Agol, Kahn--Markovic and Wise? Is the invariant field helpful to decipher the information encoded in the fundamental group?