I think the White House definitely has a part in developing a peace process for all conflicts. But how successful they are is always something that remains to be seen. Any conflict that occurs internationally is always going to be looked at by the White House as an important issue to look at.
Yes I believe the White House has developed a peace process for all conflicts. Not every conflict is the same but when trying to promote peace the talking points are always similar. No one wants to see their citizens injured or even killed. These points are discussed at a high level to lead to a peaceful compromise.
In consideration of the many battles that continue to go on across the world, it cannot be assumed that there is a peace process being developed for every conflict. The White House is not involved in every international conflict taking place. The White House likely does not develop a peace process unless parties need arbitration or if the conflict affects America directly.
No, there are conflicts that do not receive attention from the White House. One example is the ongoing civil war in Syria. While the White House did get close to urging UN sanctions on the Syrian government for attacking its own citizens, there has been no intervention in the conflict. Syria is far from the only conflict in the world to fall under the White House radar.
The White House can't even develop a peace process for all parts of Washington, it greatly overstates its influence to suggest that they're the conflict police of the planet or they even play a successful role in most conflicts they try to bring to a close. This question gives the White House far too much credit.