While I think it's lovely that you care about all three of your pets, I would personally never let them interact at all. Snakes are predators as are canines and felines. The snake may well strike at your dog or cat in a defensive manner in order to protect itself from perceived danger. It may also simply become so stressed by the exposure to the dog and cat (and the tree climbing) that it's immune system can be compromised or you may see a refusal to eat (which apparently you are seeing).

Even the most well behaved pet dog or cat can try to "play" with your snake and injure it before you can intervene. A cat often slaps at anything that moves quickly and it's not the cat's fault, it's just instinctive. Dogs often grab at things and those canine teeth or a quick shake of your dogs head can result in a severe injury to your snake.

I know I'm over cautious on some things but when it comes to snakes and other household pets, we just keep them completely apart for everybody's own good and for our peace of mind.