Why do we need to run the bellman-ford algorithm for n-1 times?
Answering your first question is simple. Try out a few graphs and see for yourself why it doesn't work. As to your second question, $n-1$ is the maximal length of a shortest path in the graph. After $k$ iterations of the Bellman–Ford algorithm, you know the minimum distance between any two vertices, when restricted to paths of length at most $k$. This is why you need $n-1$ iterations. Negative weights have absolutely nothing to do with it.

For a more sophisticated answer, consult the recent paper of Jukna and Schnitger, On the optimality of Bellman–Ford shortest path algorithm. They show that among the class of Bellman–Ford-like algorithms, the Bellman–Ford algorithm is optimal.