Can the mouths of a worm-hole be torus shaped? Theoretically, most wormholes should have spherical mouths (if they exist). Could have a wormhole have torus-shaped mouths? What about other shapes?
 A: A wormhole with a torus-shaped mouth is a perfectly well-defined spacetime, yes. Via the usual cut-and-paste construction method, you can do the following : 
Take a copy of $\Bbb R^{n-1}$, remove two non-intersecting tori $T_1, T_2$. This gives you the manifold with boundaries $$M = \Bbb R^{n-1} \setminus \left( T_1 \cup T_2 \right)$$
Define the embedding $\iota_1 : \partial T_1 \hookrightarrow M$ and $\iota_2 : \partial T_2 \hookrightarrow M$, then identify the boundary $\partial T_1$ with $\partial T_2$ with the help of the function $$\iota_2^{-1} \circ \iota_1 : \partial T_2 \to \partial T_1 $$
You obtain an appropriate spacelike hypersurface for a wormhole with a toroidal throat. You can then simply take the product of this manifold by $\Bbb R$ which, since it is non-compact, admits a Lorentz metric and is hence a proper spacetime. 
Using the Israel junction condition trick, you can form a simple thin-shell traversible wormhole by concentrating the stress-energy tensor along the boundary.
A: Mathematically, the answer is "YES".
Wormholes whose mouths have the topology of a torus are called ringholes.  
The term dates back to this 1996 paper.
The mathematical existence of these wormholes simply means that our best theory of gravity, general relativity, admits wormhole solutions to its field equations.
Physically, the answer is "UNKNOWN".
The physical existence of wormholes depends on the existence of so-called exotic matter (essentially, negative matter).  While the existence of such matter is consistent with quantum theory under certain peculiar circumstances, there is little reason to believe that stable, macroscopic wormholes (of any topology) are naturally occurring.
The general consensus is that if wormholes exist -- spherical, toroidal, or of any other topology -- they would be ephemeral, infinitesimally small objects in the vacuum state of a theory of quantum gravity.  [The creation of such a theory is a long-standing unsolved problem in theoretical physics.]
