You need a D1 or something like that. Some people use "concept" to refer to "idea", and others more narrowly use it to refer to "idea that is so important that it is conventionalized as a word (or other symbol)". I would use "concept" in the latter manner (granting that "concept" is becoming a meaningless term in current usage). There are cases of words that don't represent a single concept, such as "up". As a particle in V-particle constructions, (look up, chat up, screw up) it doesn't represent a concept, instead it's a somewhat arbitrary component of a lexical item made up of two words. You could fix this by fine-tuning the concept "word", but we don't want to beg the question.
"Made of" suggests something along the lines of language acquisition, where "animal" is "made of" the concepts "dog, cat, rat, horse...", and dog et al. are cognitively more basic (acquired earlier), and "animal" is an abstraction that subsumes a number of similar concepts. In a way, you glue concepts together. However, "husky, poodle, chihuahua" further differentiates the first-order concept "dog" so those concepts are not "made of" other concepts. But this is not a huge objection, so I think P2 is probably correct.
I will leave C1 alone for the moment, and turn to the matter of what a concept is good for. A concept, like "dog", serves a cognitive function, that it identifies an open-ended class of facts that you can perceive. A concept abstracts away from a specific instance, and allows you to identify all of those things that are of the same cognitive category, while being distinct from other things. So to have a concept, you have two or more things that are similar in a particular way yet different in irrelevant ways, and the group is distinct from some third thing.
The problem with "Buglump" is that there is no thing, much less a class of things, that are "Colorless red existing dog that is a cat and doesn't exist". The stimulus for a concept is some actual thing: it is useful to invent the concept "transistor", "syllable", "recursion" because there are real things out there that these concepts are about. There is no thing for the concept to be about; this broken concept fails to fulfill the cognitive function of a concept.
This may be what you mean when you say that the concepts of Buglump "fail to cohere and thus do not form a new idea".
There are words (especially in the political sphere) which don't actually refer to anything, rather they have an emotive function. You would not say that such words represent concepts: so it's not that all words represent concepts, rather, all concepts are represented by words.
I would suggest starting with a good grasp of what a concept is (and how it relates to ideas; what the function of a concept is, etc), for which I recommend How we know by Harry Binswanger.