Why does Jean Valjean rent the house in the Rue Plumet?
I sense two components to this question. The first would be why leave would Jean Valjean leave the protection of the convent life and need to rent the house in the first place. If Valjean does not leave, renting the house is not needed. Hence, the question of why he needs to leave can be summed up in his commitment to Cosette. Even though Valjean is quite happy living away from others in the convent, he recognizes that his paternal love towards protection might be more important than the love he holds for his own condition. Essentially, the need to leave is rooted in sacrifice for Cosette:
It will be remembered that Jean Valjean was happy in the convent... He saw Cosette every day, he felt paternity spring up and develop within him more and more, he brooded over the soul of that child... On reflecting upon this, he fell into perplexity. He interrogated himself. He asked himself if all that happiness were really his, if it were not composed of the happiness of another, of the happiness of that child which he, an old man, was confiscating and stealing; if that were not theft? He said to himself, that this child had a right to know life before renouncing it, that to deprive her in advance... was to rob a human creature of its nature and to lie to God... He resolved to quit the convent.
In making the decision to give Cosette "a right to know life before renouncing it," Valjean leaves.
The second issue is how Valjean could live in Paris away from others' prying eyes. Valjean is a wanted man and he realizes that Javert will stop at nothing to find him. It is not as if Valjean can go anywhere or simply live in Paris with Cosette. Valjean understands that being careless could endanger Cosette, and thus make life even more difficult for both of them. It is with this understanding that Valjean rents the house in the Rue Plumet. This is not just any house. It is one perfectly suited for the life that Valjean wishes to lead. The history of the house is that it was built by a high ranking Parisian chief justice who carried on with an affair and needed a place to see her that could not be seen by others. The house's appearance to other was almost sunken in and contained a labyrinth of "solitary extremity." It was here in which the house's concealed nature stood perfectly designed for someone like Valjean. While the chief justice needed concealment from the public for his own self- imposed actions, Valjean and Cosette need it for the imposed punitive nature of society. The hose at the Rue Plumet becomes a perfect and ideal choice for Valjean's desire to both live in Paris with Cosette and also remain concealed from the same elements that would necessitate Valjean's flight from it.