Translation:The boy's clothes are in that wardrobe.
In Hungarian cabinet means (a standalone) szekrény, but a built-in cabinet or wardrobe is called as beépített szekrény. Therefore, a szekrény without the adjective beépített is cabinet, with the adjective beépített together it will called wardrobe. However, beépített konyhaszekrény means in English built-in kitchen cabinet (and not wardrobe).
The first one is correct, the second one is not.
It is either "a fiúnak a ruhái" or "a fiú ruhái". There is no difference between these two. The first one is the fully formed structure, the second one is the simplified version. In some situations you can use the simple version, in others you have to use the full version.
No, it has nothing, or very little, to do with formality. It is more about grammatical necessity. Whenever you can use the simple form, use it. But, for example, the possessor-possessed structure can be broken up, the two parts separated. In such a case, you have no choice but use the full form, otherwise the structure is lost:
"A fiú ruhái abban a szekrényben vannak." - OK
"A fiúNAK abban a szekrényben vannak a ruhái." - have to use the full form. I guess you can't even do something like this in English, can you? The "boy's clothes" has to stay together. Or you have to use a different solution.
"A fiú tiszta ruhái... " - is also fine, that is just an adjective (clean) inserted, it does not break the structure. The boy's clean clothes. The same works in English, too.
There can be other cases.
"The boy's clothes" - "A fiú ruhái" - OK.
"That boy's clothes" - how do we do this?
"That boy" is "az a fiú"
"(the) boy's clothes" is "(a) fiú ruhái"
We can't just put the two together. We have to use the fully formed possessive:
"Annak a fiúnak a ruhái". - As we know, when "az a fiú" is being added some suffix, both "az" and "fiú" get the suffix: "annak a fiúnak".
There can also be some ambiguity that we want to avoid:
"a fiú kutyái" - the boy's dogs
Or is it rather "a fiú-kutyái" - "(his/her) male dogs"?
Here, "a fiúNAK A kutyái" makes it unambiguous.
"Itt van a nagymamája." - His/her grandmother is here.
"Itt van a Nagy mamája." - Nagy's mom is here. ("Nagy" being someone's family name).
We can make it clear:
"Itt van a Nagynak a mamája."
So, there you go. :)
You are welcome. Yes, it is all due to those suffixes. They sort of put a "label" on the word. You can reshuffle the words, they will still retain their suffixes and therefore their roles in the sentence. If I say "ruhái", it is already clear that I am talking about someone's possessions. Whereas if I say "clothes", it does not give me that information. The possessedness is only coded in the position of the word: after the possessor: "the boy's clothes". In English, you have no choice but keep the much stricter word order. Hungarian has more "wiggle room" in that regard. So, that "freedom" is used to express something else: emphasis, order of importance, focus.
That "has" is not so much about possession as it is about where he keeps (his) clothes, is it?
I guess we could use the same sentence, or maybe a different word order:
"A fiúnak abban a szekrényben vannak (a) ruhái."
Or we could use a different verb:
"A fiú abban a szekrényben tartja a ruháit." - The boy keeps his clothes in that cabinet.
"A fiú abban a szekrényben tart ruhákat." - The boy keeps clother in that cabinet.