Chinese words are categorized according to their grammatical functions and semantic meanings.
Unlike English, Chinese adjectives, together with intransitive and transitive verbs, belong to the verbal class. It's also important to distinguish intransitive and transitive verbs in your question.
Some Chinese words are 兼类, that is they belong to multiple word classes. It has to be a frequent usage. For those temporarily used as another words class, it's called 活用.
鸟 always behaves like a noun
This statement is incorrect. It usually behaves like a noun only, but we can indeed find widely known cases of 活用, for example, in 鸟散 it acts like an adverb.
The more I go though the dictionary, the more I get the impression that the majority of Chinese words can act both as verbs and nouns
For disyllabic verbs, we can use "object + 的 + vt./vi.", like 鲜花的盛开、自我的放飞. (Note: in this structure, vt.'s passive). 的 here cancels the independence of vt. + object or subject + vi. so that the structure can be used substantially. It's still a verb, not a noun.
The above is the generally case. However, sometimes different degrees of lexicalization could happen. In examples like 计算机的使用、水的净化, the line between verbs and nouns starts to blur.
It's hard to accept the view that intransitive verbs are widely categorized as nouns. For example, try to make a sentence with 长 as a noun. I couldn't come up with any natural one. However, as you might have noticed, it is also an adjective. It's sometimes hard to distinguish Chinese adjectives and intransitive verbs. A simple test that might not always work is: if it can be used with an adverb of degree, it's usually an adjective; if it can be followed with 着、了、过, it's an intransitive verb. In this case, 我的头发很长 and 我的头发长了 both work.
It's very common for Chinese verbs to have 兼类 but I cannot quote statistics.
since there's a consensus that Chinese distinguishes word classes, it has to be the other way around, and words behaving beyond a single word class should be more of an exception than a rule
I don't see the logic here. Why the consensus that Chinese distinguishes word classes precludes the possibility that Chinese words commonly belong to, say, two word classes? I'm not claiming it is true that Chinese words commonly belong to two word classes, just questioning the logic.
What are the consequences of using a word beyond its class boundaries in Chinese?
This is 活用. Not all cases are accepted. I don't know systematically why some work while others not, but native speakers can usually tell which is proper. A successful 活用 often makes the language more vivid, thus not commonly seen in formal texts. New usages of words sometimes are created by 活用, see 绿 in the following paragraphs.
I have a feeling that doing so in Chinese would result in completely non-understandable and unparceable sentences (mostly because of its restrictive phototactics).
My experience is no. It's very likely that I feel the same way as you feel in an English context.
if I stuck 绿色 between a subject and an object in the sense of "to make green/to paint green", would I be understood?
Yes. 我绿色了墙 is considered wrong (and should not be used) but understood as 我把墙刷成了绿色。 活用 is restrictive, in this case 绿 is more common than 绿色. 绿 is also an intransitive verb, as in 草绿了; this is 兼类，not 活用. But it's not a transitive verb, so 春风绿了芭蕉 is 活用 (here 活用 makes it very vivid and beautiful). More recently, a new usage of 绿 (vt.) has come into being，as 他被绿了, meaning he's cheated by his partner. This makes 绿 also a transitive verb （兼类）. But, if we restrict its meaning to the color, using it as a transitive verb is still a 活用, not generally acceptable but vivid when accepted.
I wonder what's the percentage of 兼类 and words that allow 活用 in Chinese. I understand that there isn't a statistic, but can you share your subjective impression?
First thing, 活用 is very restricted in two senses: not all words are allowed and the same word allowed here might not be allowed in other places. This gives me the impression that it's not common, especially in formal texts, where I would put less than 1%. Colloquially and on the internet, it's more common and I'd say 3%.
For 兼类 of noun and verb, there are some papers that count them. However, the standards are quite inconsistent. To give you a flavor, 4.8% (colloquial) and 8.43% (written) for monosyllabic words, 5.33%
(colloquial) and 27.22% (written) in disyllabic words. Other types of 兼类 are less common and less studied.
鸟散 Isn't it a compound word though?
There are a lot of words/phrases with the same structure but different degrees of lexicalization. I'd totally agree that 鸟瞰 is a word (fully lexicallized), but in terms of 鸟散, my first instinct is that it's not so tightly condensed as a word. It's not fully lexicallized probably because it's not frequently used.
Try to make a sentence with 长 as a noun Wouldn't something like "A long (one) went" work in Chinese? Or could it ever mean "longness"?
We use 长的 substantially, something like (one) is omitted after 的. For "longness", 长度 is used. 长 (cháng) on its own doesn't fulfill a substantial role.
If you responded that all adj can in fact be used both ways, they'd say that although each adjective belongs to one of the 2 classes, they can freely converse between them. Which makes the claim nonsensical because having 2 adj classes that freely converse is 100% indistinguishable from just having 1 class. ...
I see your point now.
There are large-numbered "pure" (single-classed) words in each word class, so all classes makes sense to exist. 兼类 has many types: v.+n., adj.+n., adj.+v., adj.+adv., v.+adv., v.+prep., prep.+conj., n.+cl. and etc. And also words with 2+ classes, like adj.+v.+n. 方便, n.+adj.+adv. 自然 , n.+v.+adv. 保管 and so on. Just too many to be sensible to make a word class for each of them. To make it even more complicated, sometimes one of classes is fundamental, and only some of the grammatical functions and positions of the other type can the word assume.
There are debates on 兼类 of n. and v. Some scholars think they're too many to be 兼类 and propose 名动词 for them as a separate word class. In my view it doesn't solve the problem. After all, other types of 兼类 also exist, especially adj.+v. and adj.+n., which are quite numbered.
The mainstream view still accepts 兼类.