I very nearly wrote my PhD thesis on this topic.
Here's as much as I was able to figure out, though it's hardly a direct answer to your question.
1) Say your total space is K\"ahler, and your fibers are compact. Then you can define a Levi-Civita connection on any open set consisting of smooth fibers. It turns out that this connection generates symplectomorphisms between the fibers.
2) In toric degenerations, the torus acts on the total space of the family, mostly moving them around, but preserving the zero fiber (which is why it's toric).
1+2?) Now imagine you use (1) to give a map from your general fiber $F_1$ to your special fiber $F_0$. Map further, to ${\mathfrak t}^*$, using the moment map on the toric variety.
Now you have an integrable system on $F_1$, stolen from $F_0$!
There's a problem: since $F_0$ isn't smooth, we can't actually use (1) to make the map. The hope is to take limits along the horizontal vector field to define a continuous function $F_1 \to F_0$.
3) It turns out that this is the same as following the gradient flow for the norm square of the moment map. And limits of real-analytic gradient flows on smooth varieties are well-understood, by Lojasiewicz. So if your total space is smooth, you can use this to show that the map $F_1 \to F_0$ is well-defined, continuous, and smooth away from the singularities in $F_0$.
I never got around to investigating how things change if the total space is singular (as in the Gel'fand-Cetlin-Sturmfels-Gonciulea-Lakshmibai degeneration motivating the questioner, and me too). Of course you can pick a resolution of singularities, and I guess you can
ask that the metric on the exceptional fibers be very very small, and use that to generalize Lojasiewicz' results. But I never worked on this seriously.
Example:
Let the family be $det : C^{2\times 2} \to C$. Then the $0$ fiber is the cone over $P^1 \times P^1$, so a toric variety, but the fiber over $1$ is $SL(2)$. That has a $T^2$ action, by left and right multiplication by its maximal torus, but doesn't have the rescaling action that the $0$ fiber enjoys. One can actually solve the ODE defined by the Levi-Civita/gradient flow and write down the map $SL(2) \to det^{-1}(0)$. It collapses $SU(2)$ to the singular point $0$.
What is the integrable system? Regard $SL(2)$ as $T^* S^3$, and the action variable as $(p,\vec v) \mapsto |\vec v|$. This generates unit-speed gradient flow on $T^* S^3$, which breaks down at zero vectors (the $SU(2) = S^3$) because they don't know which direction to go.
Ben is right. Consider the example of $\mathbb{C}P^1$, which is an ordinary sphere. The action of $T$ is a rotation around the vertical axis, and the orbits are latitude lines. The action of $V$ pushes the sphere from the south pole to the north pole, and the orbits are meridian lines. The symplectic form is just the area form and the action of $V$ is clearly not area-preserving.
As for your second question, at a technical level Ben is also right. Each of your "whys" is really one part of the Guillemin-Sternberg structure theorem, at least if you take the question for half-dimensional toric actions on a compact symplectic manifold. My impression is that the structure theorem is not all that easy.
But I can think of something else to say. If you take your question for toric varieties in the sense of Fulton's book, then essentially it's all true by construction. You can think of a toric variety as the answer to the question "How can we generalize the projection from the sphere to the interval, keeping all important properties?" If you start with a convex polytope, you should try to build a singular fibration over it such that the fiber over a point on a $k$-face is a $k$-torus. If it is a rational convex polytope that contains the origin, then the definition of a projective toric variety is an organized solution to this question. You build the symplectic structure and the complex algebraic structure so that it all works.
For me at least, it is helpful to first think about the case when the polytope is integrally simple, so that the variety is a manifold. Then the case when it is rationally simple, so that the variety is an orbifold.
Best Answer
As far as my understanding goes the answer is no, and I will try to explain why and clarify the list of comments (I have little reputation so I cannot comment there) and give you a partial answer. I hope I do not patronise you, since you may now already part of it.
First of all, as Torsten said, it depends what you understand for classification. In this context a torus $T$ of dimension $r$ is always an algebraic variety isomorphic to $(\mathbb{C}^*)^r$ as a group. A complex algebraic variety $X$ of finite type is toric if there exists an embedding $\iota: (\mathbb{C}^\ast)^r \hookrightarrow X$, such that the image of $\iota$ is an open set whose Zariski closure is $X$ itself and the usual multiplication in $T=\iota((\mathbb{C}^\ast)^r)$ extends to $X$ (i.e. $T$ acts on $X$).
Think about all toric varieties. It is hard to find a complete classification, i.e. being able to give the coordinates ring for each affine patch and the morphisms among them for all toric varieties.
However, when the toric varieties we consider are normal there is a structure called the fan $\Sigma$ made out of cones. All cones live in $N_\mathbb{R}\cong N\otimes \mathbb{R}$ where $N\cong \mathbb{Z}$ is a lattice. A cone is generated by several vectors of the lattices (like a high school cone, really) and a fan is a union of cones which mainly have to satisfy that they do not overlap unless the overlap is a face of the cone (another cone of smaller dimension). There is a concept of morphism of fans and hence we can speak of fans 'up to isomorphism' (elements of $\mathbf{SL}(n,\mathbb{Z})$). Given a lattice N, there is an associated torus $T_N=N\otimes (\mathbb{C}^*)$, isomorphic to the standard torus.
Then we have a 1:1 correspondence between separated normal toric varieties $X$ (which contain the torus $T_N$ as a subset) up to isomorphism and fans in $N_\mathbb{R}$ up to isomorphism. There are algorithms to compute the fan from the variety and the variety from the fan and they are not difficult at all. You can easily learn them in chapter seven of the Mirror Symmetry book, available for free. Given any toric variety (even non-normal ones) we can compute its fan, but computing back the variety of this fan many not give us the original variety unless the original is normal. You can check this easily by computing the fan of a $\mathbf{V}(x^2-y^3)$ (torus embedding $(t^3,t^2)$) which is the same as $\mathbb{C}^1$ but obviously they are not isomorphic (the former has a singularity at (0,0)). In fact, since there are only two non-isomorphic fans of dimension 1 (the one generated by $1\in \mathbb{Z}$ and the one generated by 1 and -1) we see that there are only three normal toric varieties of dimension 1, the projective line and the affine line, and the standard torus.
The proof of this statement is not easy and to be honest I have never seen it written down complete (and I would appreciate any reference if someone saw it) but I know more or less the reason, as it is explained in the book about to be published by Cox, Little and Schenck (partly available) This theorem is part of my first year report which is due by the end of September, so if you want me to send you a copy when it is finished send me an e-mail.
So, yes, in the case of normal varieties there is some 'classification' via combinatorics, but in the case of non-normal I doubt there is (I never worked with them anyways).
Become a toric fan!.