Are appeals of termination from PhD programs common and are there any unambiguously successful or unsuccessful approaches to fighting a termination?
Typically, successful appeals happen when another advisor is willing to take on the student after the appeal. This creates an ally during the appeal process. It also creates an "out" for the department. I have two friends who were able to appeal by finding an alternative advisor. I know no-one who was able to appeal with their original advisor UNLESS the advisor was actually using the process as a "wake-up call" rather than an authentic attempt to force the student out.

In the cases where an advisor is using the process as a real authentic attempt to force a student out, the advisor is communicating that they no longer desire to have you as a PhD student. The department is going to be reluctant to force that advisor to continue with you, despite the soundness of your logic. The reluctance is partially because it creates an uncomfortable/unproductive mentoring situation when one party is not enthusiastic. The reluctance is also related to an overall ethos of not wanting to oppose faculty on issues of academic judgement.

The presence of another ally/advisor who will take you on as a student after the process is concluded allows the department to evaluate your claims independent of their reluctance to force your original advisor to take you back. It allows everyone to put this down to a matter of "fit" rather than genuine problems with your work or your advisor's judgement.

IF your advisor is actually on the fence about this, you could attempt to convince your advisor that you now are "awake" to the stakes and they will find working with you to be much better/more productive in the future.