When we perceive something, we tend to categorize it. For example, when we hear the word puppy, we think of the concept of dogs and then the conceptual category of animals. Is it acceptable to claim that linguistic units are organized in conceptual categories?
Yes, there are many patterns of conceptual organization evident in language.It shows up in all sorts of psycholinguistic effects such as priming as well as in the way language structures are related to one another. One such pattern is represented in the WordNet dictionary relating synonyms, hyponyms and hypernyms in what they call synsets. There is also a pattern of conceptual organization evident in grammar - for instance verbs and nouns differentiate between thing-like and action/state-like concepts. Cases often are used in related areas - e.g. in Czech accusative denotes directionality when not used as direct object. Passive constructions are related to active constructions through the concept of agent-patient, etc.
But the key thing here is that we're not dealing with a single hierarchical structure (something like a Linnean taxonomy). Parts of language are related in all sorts of different ways. That's why grammars are so hard to write - you have to choose one pattern of organization. And it's also why it's more efficient to organize a dictionary alphabetically rather than following some sort of conceptual pattern.
Another problem I've hinted at is with what you consider a unit. Construction grammar will think of both 'rules' and 'words' as constructions and consider them as conceptual units whereas more traditional approaches may only think of 'words' as 'units' to which conceptual relationship can be applied.
But there are also many patterns of regularity in language that are clearly not conceptual (or not primarily conceptual). For instance, apart from some subtle symbolism, most phonological relationships are not conceptual. Nor are some things like patterns you only find by analyzing corpora (such as most collocations). Of course, many seeming regularities in language may just be epiphenomenal or their conceptual underpinnings may not present the same way in all speakers.
Finally, you also have to be a bit careful about what you mean by conceptual. In the above, I was drawing on ideas I think were best articulated in Lakoff's 'Women, Fire and Dangerous Things', Langacker's 'Cognitive Grammar' and Talmy's 'Cognitive Semantics'. Conceptual in this context means related to or embedded in our broader cognitive system described outside traditional approaches (Lakoff made that argument most cogently in the first 5-6 chapters of his book - the rest is just filling in the gaps).
I think that memory is more 'semantic' than 'linguistic', but, yes, we store these categories. The larger sets are 'hypernyms' of the subsets (which are 'hyponyms' of the larger sets). ex: Fido is a puppy. Every puppy is a dog. Every dog is a mammal. Every mammal is an animal. Every mammal is a living thing. (Therefore, Fido is a living thing.)
I would say that it is wrong to say that when we hear the word puppy, we think of the concept of dogs and then we think of the concept animal. That suggests a simple stimulus-response model of word recognition which I would say is wrong. Let's suppose that we have successfully heard "puppy" (not "guppy", "puffy", "putty"), then we can access whatever / wherever the mental concept "puppy" is, that is, we will have identified an intended concept. We may or not may not "think of" a specific puppy. Perhaps when you say "think of", you mean "access". We may or may not then access hierarchically related concepts like "dog", "mammal", "animal", "happiness", "paper towel" and so on. With this concept accessed, we have access to whatever other facts we know are also true of the referent.
Saying that linguistic units are organized in conceptual categories implies that the categories exist independent of their symbolic labels, thus the categories "green", "blue" and "grue" (which is the unification of "green" and "blue") exist, even if your language does not have any of these categories. Presumably you see the problem with that. It would, however, be correct to say that when a concept is formed, it needs some (linguistic) label so that you can use the concept. From the standpoint of acquisition, it is most likely that a person will first learn a word and then learn what it refers to, but adults especially can know of real-world things without knowing what they are called (for example "soffit"). In that intermediate period, you can have an awareness that there are these things on / in houses, but you don't know the word "soffit". In that case, I would conclude that you have an incomplete concept, lacking the label that allows you to exploit the concept.
One of the other sides of the coin is the question whether each linguistic unit does represent a conceptual category. It is easy to see that that is not always so, for example with "dog". That refers to at least three distinct concepts. One is a concept of action, of "pestering". One refers to a species of canid; and one refers to a thing that blocks movement (e.g. "feed dogs" on a sewing machine).
No. But this sort of mentalism is more empty and useless than it is wrong.
Suppose I try to explain the meaning of your question by saying it consists of the concept of "when", combined with the concept of "we", combined with the concept of "perceive", and so on, no one will be the wiser about what you've said. Such an account is useless. As to whether it's correct, well, who knows? Who cares?
If you give such an account supplemented with an explicit version of "combine with", so that you can tell exactly when concepts can combine to give a sensible result, you get the sort of "semantic" theory called Montague Grammar, or Montague Semantics (after Richard Montague who pioneered such theories), and then opinions will differ about whether you're accomplishing anything beyond giving a syntax. I doubt it.