Mrs Pumphrey was worried about Tricki because he would not eat anything. He refused to eat his favourite dishes and had bouts of vomiting. He spent all the time lying on the rug and panting. 

She called Mr. Herriot, the veterinary surgeon to help Tricki. Yes, she made a wise decision. 

‘I’ refers to the veterinary surgeon, Mr Herriot. He has narrated the whole story.

No, the narrator wasn't as rich as Tricki's mistress. It is clearly evident from the privileges that Tricki enjoys.