"كُرْسي الْقِطّة غَريب."
Translation:The cat's chair is weird.
Because in the Arabic sentence above, the word غريب (strange/weird) is an adjective that has been used as a predicative (telling information) and not attributively to describe the cat's chair.
Your sentence would rather be: كرسي القطة الغريب (notice how the adjective now gets AL to be defined just like the noun composite كرسي القطة - these are 2 words bound together by a genitive relation (the chair of the cat/ the cat's chair) and since it is defined and I want to describe it, then the adjective has to be defined like the noun (attributive adjective follows the noun in its cases, gender, definition, and number). This is not the case above however, because the adjective had been used to TELL information (you can tell that by noticing the "is" in the English translation).
As I understand it, your sentence says (the weird chair of the cats). If this is the case, that would be: كرسي القطط الغريب (kursiyyu al-QiTaTi al-ğaríb). Before explaining further the grammatical situation here I just need a confirmation if this is what you meant in English.
So to sum up the possible combinations from this thread:
"the weird cat's chair" = كرسي القطة الغريبة (the cat is weird, adjective describing the cat is feminine)
"the cat's weird chair" = كرسي القطة الغريب (the chair is weird, adjective describing the chair is masculine)
"the cats' weird chair" = كرسي القطط الغريب (the chair is weird, adjective describing the chair is masculine)
My conclusion: in the examples above, we can tell which part of the Iḍāfah is modified by the adjective, because the possession and the possessor vary in gender. But a phrase like كرسي الكلب الغريب must be ambiguous by nature - we have no way of telling whether it's "the dog's weird chair" or "the weird dog's chair"? Is it correct???
For the last example, the knowledge is somewhat mutual between the speaker and the listener depending on the context. Unless other stuff are added to make it less ambiguous, e.g. الكرسي الغريب للكلب (the weird chair for the dog) - this is one of the ways that we can express possessive attribute; by using the preposition لِـ (to/for). I would say though that كرسي الكلب الغريب would initially be understood or automatically somehow understood, right away, that الغريب is an adjective to الكلب since they follow each other; I guess how our brains work right away by attaching successive words together unless the context says something else.
One more option also we can do here is by changing the nature of the sentence (or should I say the phrase). If I want to emphasize the fact that the chair is weird, then we might say كرسي الكلب هذا غريب (this dog chair is weird). Of course we changed the structure here completely and it is a full sentence instead of just a phrase composed of nouns+adjective.
I would say it is probably that way in most languages that if a speaker fears some ambiguity, then the speaker would have to re-arrange the words and change the structure of the sentence to deliver the idea better.
Most welcome. One thing I didn't mention though, is that, in all these patterns above, it is taken as a separate entity or a block of its own without context.
Meanings can be clear (and the adjective would be clear to which it belongs) if the block is within a context. This is because when a Genitive compound falls in a place of accusative for example, the first noun would be Mansoob (i.e. ending with fatHa or -a) but the second noun remains as it is as the genitive case requires Kasrah (-i). Hence, when the adjective comes afterward and it ends with fatHa -a, we would know it refers back to the first noun, but if it ends with kasrah (-i) it refers to the second noun in the genitive. This is because attributive adjectives (those attached to the nouns) resemble and follow the attributes of the noun (unlike predicative adjectives which act on their own because they tell some information, i.e. act as predicate).
This is in a nutshell and thought I should mention it here to keep things in perspective but going further might get a bit tricky and complicated.
In Arabic, nominal sentences (sentences starting with a noun) mostly start with a definite noun. Thus, sentences like a cat's chair is weird are not quite possible and must be forced into the cat's chair is weird. There are few exceptions but those are quite "deep" in grammar and literature and not the regular normal use. Sometimes also, the context might play a role when translating into Arabic, and expressions or sentences are not translated one-to-one.
I guess by "possessive" you mean the Genitive case or the "OF-relation" between two nouns.
In Arabic, there is no "of" in that sense (unlike dialects nowadays). The nouns are simply added together (following each other): the chair of the woman would be something like chair the woman كرسي المرأة, Notice here how the definite article (if it exists) it is added to the SECOND term in the compound (which is here, in our case, Al-Mar2ah المرأة) - so unlike English where you would say THE chair of THE woman, this structure doesn't happen in Arabic Genitive.
The Genitive case requires also adding "kasrah" (that is -i) sound to the second term (we say the noun is majroor). This is in the case of simple nouns; Other cases do have other signs. Thus, to be complete, the chair of the woman would be كُرْسِيُّ المَرْأَةِ (kursiyyu Al-mar2ati) [or if you like it more phonetically: kursiyyuL-mar2ati].
side note: Genitive is called Idhaafah إضافة in Arabic, meaning "addition" because words are simply added together.