**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 your second question is **no**.  For example, the biadjacency matrix of the complete bipartite graph $K_{n,n}$ obviously cannot be permuted to a lower triangular matrix, but this graph has many perfect matchings.  A necessary and sufficient condition for the graph to have zero perfect matchings is to fail Hall's condition, which you can say in terms of the biadjacency matrix (if you like). 


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