Sarah, your story and reactions make much sense to me.
~~#F00:"I have had rashes all the time, day and night, ever since I remember, all different body parts, inside and out."~~
That means your sensitization for contact/separation issues already took place in early childhood, at birth, or even before that. Do you know about the conditions of your birth?
Since then, you have reacted with many skin areas based on that sensitization - this can be due to a variety of conflicts, but also due to exposure  for sun, chemicals etc. Your skin is hypervigilant, like a sympathetic response, because its baseline calm and resilience is reduced.

~~#F00:"My cat Felix means EVERYTHING to me!"~~
He is like your child and you believe that you alone are responsible for his life and wellbeing. You had 10 years of nightmares after 3 cats had disappeared, you imagined in those dreams that they had starved. The disappearances AND the dreams were traumatic and imprinted you with an exaggerated sense of responsibility, caring, nurturing and protecting your relations. (This is a so-called "breast gland constellation" according to GNM, we'll speak about that later in the course)
Such a blueprint predestines you for separation conflicts, too...

~~#F00:Every day I was fighting like crazy to find people that help search, to get his travel bag and paperwork from the people that blocked my number.
17th of November inside of the thighs - very itchy ~~
The inside of arms and legs are usually about holding on, and that's what you had been trying to do for weeks. Itchy rashes often appear after a resolution. Directly before, what might have happened that gave your legs this impression? Did you wake up with the itch, or did it start during the day?
Same applies for the 4th december.

There was a big resolution when you heard of Felix, and another one when you learned that he was fine. Swellings are "normal" at those times when we had such emotional burdens that loosen. Your face reacts because you come closer to Felix, he is getting almost into reach, your skin anticipates to rub and kiss him... and that's where you are at.

It will resolve, but in order to re-train your skin, we'd also have to look onto your own abandonment, almost starving, and neglect of care. I'd guess that you re-enact your history? 
For now, I'm  convinced that tapping with predominant current feelings and thoughts around Felix and yourself, one at a time, will help your skin to recover!.