Answers: (i) Yes, if we replace states by weights (not every
von Neumann algebra admits a faithful state);
(ii) Yes (for all von Neumann algebras); (iii) All of them.

Suppose M is an arbitrary von Neumann algebra and p≥0 is a real number.
Then we define a right L_p(M)-module as a right M-module equipped
with an inner product with values in L_{2p}(M), satisfying the same algebraic
properties as for Hilbert W*-modules together with the appropriate completeness
condition (we require completeness in the measurable topology,
which coincides with the σ-weak topology for p=0 and with the norm topology for p>0).
Here L_p(M)=L^{1/p} denotes the L_p-space of M, in particular,
L_0(M)=L^∞(M)=M, L_1(M)=L^1(M)=M_* (the predual), L_{1/2}(M)=L^2(M)=the Hilbert
space of half-densities on M. (The subscript notation is much more natural than the superscript notation
because L_p-spaces form a graded algebra, p being the grading.)
A morphism of right L_p(M)-modules is defined as a morphism of algebraic right M-modules
that is continuous in the measurable topology.
It turns out that right L_p(M)-modules form a W*-category.

We observe that the category of representations of M on Hilbert spaces is equivalent
to the category of right L_{1/2}(M)-modules.
If we have a right L_{1/2}(M)-module X with an inner product x,y↦(x,y)∈L_1(M),
then x,y↦tr(x,y)∈C is a complex-valued inner product on X,
which turns X into a Hilbert space together with an action of M.
Vice versa, if X is a Hilbert space equipped with an action of M,
then x,y→(w∈M↦(x,yw)∈C)∈L_1(M) is the corresponding L_1(M)-valued inner product.

Suppose 0≤p≤q are real numbers. We define a functor from the category
of right L_p(M)-modules to the category of right L_q(M)-modules
by sending a right L_p(M)-module X to X⊗L_{q-p}(M).
Here ⊗ denotes the algebraic tensor product, without any kind of completion.
Although it is non-obvious, in the end this tensor product turns out to be complete.
Likewise, we define a functor from the category of right L_q(M)-modules
to the category of right L_p(M)-modules by sending a right L_q(M)-module Y
to Hom_M(L_{q-p}(M),Y).
Here Hom_M denotes the space of algebraic homomorphisms preserving the right action of M,
without any kind of continuity property.
Again it is a non-obvious fact that this space is actually a right L_p(M)-module.
One can prove that the two functors defined above form an adjoint unitary equivalence
of the W*-categories of right L_p(M) and L_q(M) modules.

In particular, the category of Hilbert W*-modules over M and the category
of representations of M on Hilbert spaces are equivalent.

The result above extends to bimodules.
An M-L_p(N)-bimodule is a right L_p(N)-module X together
with a morphism of von Neumann algebras A→End_N(X). (The algebra
of endomorphisms of any object in a W*-category is a von Neumann algebra.)
Since the above equivalence is an equivalence of W*-categories,
we can immediately extend it to an equivalence of categories of M-L_p(N) and M-L_q(N) bimodules.

In particular, the category of Hilbert W*-bimodules from M to N is equivalent to the category
of Connes' correspondences from M to N.

Moreover, one can observe that the bicategory of von Neumann algebras, Connes' correspondences, which compose via Connes' fusion, and their intertwiners
is equivalent to the bicategory of von Neumann algebras, Hilbert W*-bimodules,
which compose via the completed tensor product, and their intertwiners.
This result is also valid for arbitrary p.

References:

The equivalence in the last paragraph of the answer was apparently first proven by Baillet, Denizeau, and Havet in their 1988 paper Indice d'une espérance conditionnelle.

L_p(M) modules were defined by Junge and Sherman in their 2005 paper Noncommutative L^p modules.

I am not aware of any paper that proves the above equivalences for arbitrary p, but I will include a proof of these statements in my thesis.