Since your Jester is still in the process of adjusting to her new cage and surroundings and given the fact she flies when scared and bumps into things in the process, it's best to not yet place the pedi perch in the cage and only do so when she is fully settled in and is completely comfortable and at ease on the cage and around you.
During this time, you can cover her cage on 3 sides (leaving front part uncovered) and have some soothing music on for her. This doesn't necessarily have to be budgie sounds and since she is a solo budgie, later on she may wonder where the singing budgie friends are and become agitated and anxious to meet them.
As for the cats, it's extremely important to keep them away from Jester, even when she is inside the cage.
The room Jester is in should be a completely cat free room.
There is no need for you to get Jester a nest box or any other type of hiding spot on the cage. This would only trigger breeding behaviour, leading to unwanted egg laying (even for a solo female budgie): https://talkbudgies.com/budgie-breedi...want-eggs.html
Yes, in order to encourage Jester to try out fresh foods, you can sprinkle a few seeds onto a leafy green, for example.
For all the detailed information and tips on how to improve your budgie's diet, be sure to check the Diet and Nutrition section.
When you offered Jester a bath, she most likely perched on it and since the tail feathers are the biggest ones in size, she inadvertently dunked them onto the bowl/birdie bath.
Once she feels the need to bathe, she will do so and the tail feathers will not be the only ones that get wet.