I've heard that cat shits in weird places can be a territorial thing. It may well be a problem caused by the cat being in a new location, that goes away by itself after the cat has had enough time to spread it's odour about the house.
My siamese started pooping in random places when she was really sick....
and about the purring, it could mean many things. Another cat of mine was going through a really rough medical condition a few years back and i stayed by her through all the process. She was almost unable to breath, and yet, she purred like never before.
She was given to me by her former owner who could no longer care for her, as she had adopted a rather aggressive pit bull who had picked up and thrown this cat. Rather disappointing to give up the cat she had loved for twelve years in favor of a dog but whatever. AFAIK there are no health issues, she seems pretty spry and didn't start purring until she was clearly quite happy.
It sounds crazy, but the cat might be fucking pissed about this abandonment and acting out. Cats hate change and this cat just went through at least two big ones: its home being invaded by the dog and then being ousted and thrust into a new completely foreign environment. I'd say try changing the litter, all the better if you can use the same litter the previous owner used to buy, try to get the same food it is used to as well, and give it a little time and some love and hopefully it'll come around and stop.
Also yes I am judging the previous owner, that is just incorrigible.
It saddens me that someone can do something like that. A neighbor recently adopted a Carey that was dumped by her +10 years former family because they were going to live abroad. She is adapting herself, but still visibly affected.
About the pooping, maybe it is just a learning issue. When i rescued my tortie from the streets, she started pooping not randomly but inside (or pretty close if the door was closed) of my bathroom. I just took her to another room alone (she is pretty hyperactive, 3 mins was enough to get her bored). After the second time, she naver did it again, and now happily uses the sandbox.
Someone told me that cats dont understand reprimands but they surely understand being bored. I dont know if that is true, but it helped me.
Edit: cat tax. Gusfraba sleeping in the geometrical center of my bed, because she obviously needs all the space