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.
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.