"dies" is a verb (the nouns "die" or "dies" would make no sense and there has to be a non-aux verb in the second half of the sentence). However, it seems to me more appropriate to say " ... that your dog is dying" or "... that your your has died." as you suggest. Maybe, "dying" is no good use of English in this case as e.g. "loving".
I wrote: "Some people think that Horatio WERE is 1 an alcoholic, but because he is so resistant to the effects of alcohol, he rarely gets drunk. ..."
Gymlish correction says that is false " 'IS' is the proper conjugation of the verb 'to be' in the present tense third person singular form. This is the best choice here"
I don't get it. Horatio is NOT an alcoholic, and the speaker does know that. So why not "WERE" (or WAS)?