"I guess you're sad that your dog has dies".
Is "dies" a Nomen or a verb?
When it is a nomen what does it mean?
When it is a verb it must be "has died", right?
"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".