@Tunamelt re food - any food changes have to be gradual. As in add 25% new food, 75% old food for a week. Then 50/50, then 75/25.
In terms of food, I feed a balanced kibble with human grade raw meat. With my westie, he can't tolerate high fat food eg regular beef mince - it gives him an upset tum. He gets very lean beef mince, chicken or turkey. Turkey is super lean.
With the chicken and rice that you need to feed her, break up the chicken and thoroughly mix through the rice so she can't avoid the rice. Cooked pumpkin can also help tums. Just boil peeled pumpkin and mash up and mix in with the chicken and rice.
You sound inexperienced with dogs. A clear warning that the "rescue" you got the dog from was crap is the fact the dog came with cat kibble as part of its daily food. Dogs cannot eat cat food as it is too high in protein and fat. The dog should have been on a balanced, complete dog kibble. The cat food could very well be the cause of the diarrhoea as it can cause pancreatitis in dogs - did you tell the vet the dog was regularly fed cat food? Even if it isn't pancreatitis, the fat content of cat food would majorly upset my westie's tum.
In terms of the dog following you around, mine always like to be where I am. They go off and do their own thing but then check in regularly and typically sleep in the office when I work from home.
What are the underlying breeds of the dog? That's important to know as breeds can have innate behaviours eg border collies herd, retrievers retrieve, terriers (my favourites) terrorise LOL Additionally breeds can have consistent temperaments eg livestock guardians are aloof and not lap dogs, some breeds are known for stubbornness. Westie owners affectionately note their "westitude".
Understanding what breeds make up your dog can help you to know what activites they will respond best to.
Personally I'd be handing the dog back to the rescue as you will have the next decade plus of deling with these behaviours. They will never go away.