Does any smooth hypersurface in (C^*)^n admit a smooth normal crossings compactifcation as a hypersurface in a toric variety? Question 1. Given a Laurent polynomial $f(z_1,\cdots,z_n)$, such that the corresponding zero locus $Z$ of $f$ in $(\mathbb{C}^*)^n$ is smooth, can we find a smooth toric variety $\bar{X}$  (together with a line bundle $L$ and an extension of $f$) such that:
a) the compactified hypersurface $\bar{Z}$ is smooth. 
b) let $D$ be the toric divisors, $\bar{Z} \cap D$ is smooth normal crossings.
More precise question: What conditions are needed to ensure that $\bar{Z} \cap D$ is ample? What is the algorithm for producing these compactifications?
 A: An obvious thing to try is to consider the Newton polytope $\Delta$ of $f$ and take $\rm Proj$ of the corresponding semigroup algebra
$$
P=\rm{Proj}\oplus_{k\geq 0}\mathbb C[k\Delta].
$$
Then $Z$ is $\rm{Proj}\oplus_{k\geq 0}\mathbb C[k\Delta]/\langle f\rangle$.
You are not a priori guaranteed that the intersection of $Z$ with the smaller dimensional strata are transversal (although that would be the case if you wiggled the coefficients a bit). The condition for that is smoothness of the hypersurfaces given by a restriction to
a face of $\Delta$. See Batyrev's paper "Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori" for this notion of non-degeneracy.
You are definitely not guaranteed that that $P$ is smooth, but perhaps this is what you are trying to look for. Attempts at toric desingularization will destroy ampleness (but you will still have base point free and big properties). 
If $\Delta$ is simple (the number of facets through each vertex is equal to the dimension) then $P$ has only abelian quotient singularities. In that case you can 
consider the corresponding smooth toric stack, and you would still have ampleness. 
If you really insist on the best case scenario, i.e. $P$ is a smooth variety, then you want to require the simplicity of $\Delta$ as well as unimodularity, which means that the aforementioned facets through each vertex come from pairing with an \emph{integer} basis of the dual lattice.
