1. You have already appealed. That was your petition for readmission, and it was denied. It's unfortunate that one of the team members gave you false hope, but usually this kind of thing is a team decision, not an individual decision.
2. If you review the sections your student handbook on cheating and plagiarism, you will almost certainly find that both are grounds for expulsion - permanent removal from the program, or even college/university as a whole. If a school enforces its student handbook, there's no grievance. Nurses must be of strong character and trustworthy; that's why even licensed nurses can face Board repercussions or lose their licenses when they demonstrate poor moral character, even if they are clinically strong. Ultimately, you stole another student's work and presented it as your own, which is cheating and plagiarism no matter how you look at it or why you did it. It can't be tolerated in professional healthcare programs because of a) character, as mentioned above, and b) that kind of practice allows students who don't understand the material to pass courses, which is a patient safety issue. Procrastination and time management are not an excuse, so I really admire that you took accountability for your poor ethical decision. That must have been hard to do, but you should be proud of yourself for telling the truth. Nurses are faced with deadlines and time management issues daily. Imagine the repercussions if they started documenting uncompleted tasks as done or just making up assessments because they got overwhelmed by the time pressures.
3. Did the Foley incident occur in the same course as the plagiarism incident? If that was the clinical associated with that lecture course, it might explain why the plagiarism is reflected in your clinical eval. I can understand why it's frustrating either way, but things are often connected in nursing school.
4. Never never tell a patient it's your first time to perform a skill on a patient. That isn't therapeutic and probably induces anxiety, even though you got a nice patient in this case. Tell the nurse you'll be working with, but do so away from the patient's bedside. If you get accepted into a different nursing school, verbally walk through the steps with your instructor/assigned nurse before entering the room. In the room, show confidence in the skill. It sounds like you were just trying to make small talk to make things more comfortable, but hearing you ask procedural questions could easily make the nurse and patient uneasy. If the patient got a CAUTI a few days later, you can bet she'd be thinking "well yeah, they let a student who didn't know anything about Foleys insert mine."
I think you've run your course with this program, as hard as that is to hear. Unfortunately, it might get worse before it gets better as you seek acceptance in other programs after this dismissal. This may change the path of your journey, but it's not the end of your journey. I wish you the best in your future endeavors.