Polynomial algorithms for UPB (Unextendable Product Bases)
I'm a little baffled by question (1). An unextendable product basis exists in $H_1 \otimes H_2 \otimes \ldots \otimes H_n$ if $n\geq 3$ or if $n=2$ and $\dim H_1, \dim H_2 \geq 3$. In all of these cases, it is straightforward to find one. 

For question (2), the question is equivalent to checking whether there is a tensor product state in the subspace which is the complement of the space spanned by the basis. Leonid Gurvits has shown that checking whether a general subspace contains a tensor product state is NP-hard, so I suspect that it is hard in this case as well.