The answer to Question I is **yes**. It is true that if a bipartite graph has a unique perfect matching, then the biadjacency matrix can be permuted to be lower triangular.  This was proved by Chris Godsil in Lemma 2.1 of the paper [*Inverses of trees*][1].  

The answer to Question II is **no**.  For example, the biadjacency matrix of the complete bipartite graph $K_{n,n}$ together with two isolated vertices cannot be permuted to a lower triangular matrix, but this graph has zero perfect matchings.  

For Question III, one option is to compute a maximum matching (this can be done in polynomial time).  If the maximum matching is not a perfect matching, then the graph has zero perfect matchings.  If the maximum matching is a perfect matching, then you can easily test if it is unique (in polynomial time) using Godsil's theorem above.  


  [1]: https://link.springer.com/article/10.1007/BF02579440