Say Goodbye to Felix - It's Dragon Talk

I was recently discussing with two friends, as one does, the important matter of whether a human being could communicate with a red dragon. Could a human learn a dragon's language? And could a red dragon learn a human's? More broadly, is cross-species communication in a fantasy world possible at all? 

This is both a more simple and a more complicated question than it appears. 

First, clearly cross-species communication is possible in the world which we live in. You communicate with your dog. You may even communicate with your cat - who the hell knows with them, though? Famers communicate with their cattle and sheep. Honeyguides communicate with hunter-gatherers in Africa. And so on. Clearly, the appropriate question is not whether some communication is possible between a human and a red dragon (or an orc and an elven cat, or a duergar and a desert troll, or an ogre mage and a tabaxi, or...), but whether and to what extent that communication can take place through speech.

Second, the issue may be an irrelevance if the setting operates on fairy tale logic. Nobody in a fairy tale fails to communicate with anybody else. Wherever they go, and whoever they encounter, the main character(s) can understand and be understood.

Third, there are lots of nuances. We might want to interrogate:

• The Chewbacca Problem. Chewbacca can understand what is said to him, and Han Solo can understand Chewbacca. But the two are not physically equipped to make the relevant sounds in each other's languages. That works in Star Wars, but is it plausible? Are we satisfied that it is possible to make sense of sounds linguistically if we are not able at least in theory to vocalise them? My feeling is that the answer to this question is yes, given enough time for the ear to become accustomed to those sounds, but I have no reason to assume this beyond gut feeling.
• The Conceptual Problem. Languages are not just made up of nouns. They are figurative and rooted in feelings and emotions. Consider: 'I hope it doesn't rain tomorrow.' Would that sentence make sense to a creature that does not have a concept of 'hope', such as an orc or kuo-toa may indeed not? What concepts to do orcs or kuo-toa have that we lack? 
• The Inferential Problem. Languages rely on inference, and cultural embeddedness, to work. Consider: 'I've been sitting around waiting for a response to my email for months.' Have you really been literally sitting around waiting for a response, or have you just been waiting? We all know that the turn of phrase is figurative. What about creatures which literally do not sit (because they have a snake tail, like a naga) or which do not sit when waiting? And what kind of inferences would be necessary to understand the speech of dragons, or bullywugs, or ixitxachitl? 
• The Embodiment Problem. Languages are spoken by physical beings with bodies, and this affects not just the act of speaking (we have mouths that can only produce a limited range of sounds) but also how we express ourselves. Consider 'I have a good grasp of X now.' To an animal which does grasp things, that makes sense. What about to one which doesn't? Does an elven cat understand 'grasping' and why that would mean 'understanding'? 

The interesting question for me is whether a 'common tongue' is a remotely plausible concept. Clearly, it is useful to make a game work. But how confident are we that humans, orcs, elves, dragons, giants, trolls, wraiths, demons and locathah would share enough thought processes to be able to come up with common language that contained anything other than simple nouns? It seems likely that all of those creatures could find agreement on things like what a rock is, what grass is, what death is, and so on. But would nuanced communication be achievable at all?