With Perthe's disease I think we have a little problem since it is a form of necrosis of the bone (caput femoris) Encouraging training would put extra pressure on the site and depending on how strong the caput is and the load on the joinmt it might effect the joint in the future so as long as it is active I would not load the joint to much with exercise like running jumping. Having said that I assume that exercise which does not have a big impact on the joint should be alright also because it would stimulate an overall development. Exercise could involve swimming which will not put a great pressure on the joints while it gives the option to train muscles so they will support the joint.
About difference in leg length, in general 1.5cM is acceptable of not causing to much trouble regarding loading other joints but the highest load will be on the shortest leg. One can't win can one?
Cheers