First off, I LOVE both of these questions...
Dragons can eat and digest almost anything, up to an including rocks. Because they are living batteries of elemental power, they don't need to eat to live. However, they do need to eat to grow. So, a dragon couldn't starve to death, but lack of eating over a long period of time would result in less growth. This is why very old dragons (10,000+ years) are much bigger now that younger dragons will be at that same age: they used to be able to stay in dragon form all the time and eat as much as they wished. Modern dragons have to remain in human form and eat considerably less. In human form, dragons can only digest regular food.
However, even though dragons can eat inorganic matter like furniture and old tires, it doesn't mean they will, or that they like it. The more inorganic something is, the less sustenance they draw from it so the more of it they have to eat to thrive. I don't know the exact balance, but that would be like thirty old lawn-mowers to one live sheep or something. Also, remember that a lot of things we consider inorganic are actually made of decomposed or calcified organic matter, like some rocks, wooden furniture, or oil. So yes, a dragon could drink oil, eat all your furniture and then have a go at your model airplanes for the glue...but they wouldn't like it very much. Most modern dragons eat for pleasure, rather than survival, so this isn't an issue.
As for dealing with waste...that depends on their form. In human form, they process waste normally. This is why Garrett has bathrooms in his lair. Dragons in human form tend to eat for pleasure, they don't have to do it for survival. Their bodies are still sustained by their internal elemental conduit, though it is closed off a bit in the smaller form so they don't have access to their breath attack. In dragon form, the elemental conduit consumes any waste, so that isn't a problem for dragons. In other words: they don't have to go to the bathroom. What nourishment they get from whatever they consume goes to their bodies. The rest is obliterated by their elemental power. When they change from human to dragon form, any undigested waste in their bodies is similarly destroyed by the reconnection to the full elemental conduit.
I know what some of you are thinking: that doesn't make sense biologically...you're right. It doesn't. That's what makes it fantasy. I couldn't have archaeologists stumbling onto the remains of dragons--bones, calcified waste, fossils, etc--otherwise their existence wouldn't be a secret. I suppose I could have had the various government agencies that deal with otherworldly matters attempt to cover or suppress any archaeological find, but I have a hard time believing that would work for long. Eventually, some farm boy in France or desert nomad in Egypt would stumble on a dragon-bone sticking up out of the earth or a big piece of fossilized dragon crap in the sand and the make a YouTube video out of it for the whole world. In my books, very few people know that things like vampires and werebeasts exist. Even fewer know about dragons.