My name is XXXXX XXXXX I have been a Vet Tech for 30+ years. This is a very interesting question you have posed. Animals do communicate with each other in different ways. Cats and Dogs will communicate with vocalizations as well as body language. In the Reptile world lizards will communicate not with vocalizations, but with body movements.
A male lizard will stomp its feet to show dominance over another lizard. They will head bobb to get anothers attention, some lizards will wave.
Bottle Nose Dolphins and Whales, have recently been found to have their own language which consists of squeaks and squeals which they are able to hold a conversation. Apes and monkeys to have vocal conversations between each other. So to answer your question, Animals of different species, do hold conversations with each other, but it may not be exactly as humans do, but they do communicate with each other to get complete thoughts across. I hope this information is helpful. My goal is to give you 100% Excellent Service.
This is an example of what I have seen in my own animals: My dog will communicate with the dogs across the street. It seems that there were some kids teasing the two dogs across the street. My dog did not see the incident, but yet when those same kids came by my home, my dog became very defensive. The key that makes me believe she had communicated was she had seen those kids several times and totally ignored them, until after she had been out and was barking back and forth with the two dogs across the street. Then later that day the incident with her and the kids occurred.
As far as sympathizing, I have seen cats and dogs care for each other when not feeling well. They also grieve when they loose a companion or an owner. Animals do communicate with each other. We may not understand their language, but they certainly understand each other. Joan