Other animals certainly appear to experience most of the same emotional states we do. We can't know for sure about things like nostalgia & sentimentality, though, without stepping into their brains. A large part of our emotional constituency arises from our use of language - animals have their own way of thinking, but our thoughts are generally tied to words, and that alters their impact somewhat.
So animals, imho, are more sensitive to their emotions than we are. That's why they're so much more reactive to them - they don't have "id" in their head to tell them to supress those emotions. We drown out our emotions with internal chatter, words upon words upon words. Non-linguistic animals can't do that, and so their internal states are more... "pure", I guess would be the proper word.