"Today is an important day for her because today is her birthday!"
Translation:Bugün onun için önemli bir gün çünkü bugün onun doğum günü!
I think that this is not the case. I read about chained compount nouns: "evin kapısının penceresi. the house's door's window. And with various plural forms: evin kapısının pencereleri. the house's door's windows. evlerin kapısının penceresi. the houses' door's window. As you can see each "chained" possessor takes both the possessed suffix in -i and the ownwership suffix in -in [hence kapı-sı -nın door-the-of] , and the possessed item suffix is -i [hence pencere-si window-the (Sing.) or pencere-ler-i windows-the (Plural] " So, I think that it is not explained! Any ideas?
Let me break it down a bit
Ev - House
"Ev"in kapısı - The door of the house
"Evinin kapısı"nın penceresi - The window of the door of the house
"Evin kapısının penceresi"nin camı - The glass of the window of the door of the house
So you break down a compound noun into a possessor and a possessed item. The possessor takes -In whereas the possessed takes (s)I. If it's a chained noun it means the possessor consists of another compound noun which I showed with quotation marks.
As for your initial question:
O - He
Doğum günü - Birthday
Onun doğum günü - His birthday
In this case doğum günü is an indefinite compound noun and the possessor (doğum) doesn't take any suffix.
It wouldn't be onun doğum gününü unless it is in the accusative form. -(n)I is the accusative ending whereas the possessor suffix is -(s)I. Normally it should have been Onun doğum günüsü but it's wrong because the possessor suffix cannot be repeated for some reason. So it's just onun doğum günü.
My another theory is we take 'onun doğum' as a possessor and günü as a possessed item. So it becomes like:
O -> He
Doğum -> Birth
"Doğum" günü -> Birthday
"Onun doğum" günü -> His birthday
You cannot say 'Onun doğumu günü'. because doğum günü is like a single word as birthday and you cannot break it by adding a suffix into the possessor (doğum).
I hope it's clear now. Let me know if you are still confused.
Short answer: No :-)
Long answer: "var" means "there is/are" or "exists" (or sometimes translates as "have/has"). So it is unnecessary/incorrect in this case.
"günüsü" is also incorrect (although it's a logical conclusion). As orde90 pointed out, "onun doğum günü" is a chained compound noun (zincirleme isim tamlaması). In these cases, we are already using the 3rd person possessive suffix, and we we don't repeat it. If necessary, we could replace it with 1st or 2nd person possessive: doğum günüm (my birthday); doğum günün /gününuz (your birthday) :-)
(I'm not a native speaker, just a grammar-nerd... I'm always open to correction, but this is my understanding. Hope it helps!)