Learn Hardcore Turkish: Hediye kartını doğum gününde masaya bıraktım, çok sevindi. - I left the gift card on the table on her birthday; she was very happy.

Because it’s a definite direct object, so it takes the accusative. The base expression is the compound hediye kartı (gift card). When you add the accusative to a vowel-final form like kartı, you insert the buffer n:

hediye kartı + (Acc) -nı → hediye kartını Without the accusative, it would be indefinite: hediye kartı bıraktım = I left a/one gift card (not a specific one).

Does hediye kartını mean “his/her gift card”?

Not necessarily. In noun compounds like hediye kartı, the -ı on kartı is a compound/possessive marker required by the grammar of noun–noun compounds, not actual ownership. So hediye kartını most naturally means “the gift card” (definite). If you need to make ownership explicit, add a genitive possessor:

onun hediye kartını = his/her gift card (definite object)

What exactly is going on in doğum gününde? Does it mean “on his/her birthday”?

Breakdown:

doğum günü = birthday (a noun–noun compound; gün-ü carries the compound/3sg poss marker)
Add locative -(d)a/(d)e to a vowel-final possessed form → insert buffer n: günü + (Loc) -nde → gününde So doğum gününde = “on (someone’s) birthday.” It’s ambiguous between “on his/her birthday” and “on your birthday” in writing; context disambiguates. To make it explicit:
Onun doğum gününde = on his/her birthday
Benim doğum günümde = on my birthday
Senin doğum gününde = on your birthday

Why is it masaya and not masada?

masaya is dative (-a/-e) and indicates motion/goal: onto/to the table. It fits verbs of placing like bırakmak/koymak.
masada is locative (-da/-de) and means “on the table” (location, not motion). With a verb like bırakmak, you typically don’t use locative to express the act of placing. You can also say the more explicit “onto the top of the table”:
masanın üstüne/üzerine (bıraktım/koydum)

Why is it bıraktım and not something like bırakdım?

Past tense -DI assimilates to -tı/-ti after a voiceless consonant like k. Then add 1sg -m with vowel harmony:

bırak- + -tı + -m → bıraktım

Is the comma between the clauses okay in Turkish?

Yes. Turkish often links two closely related clauses with a comma: …, çok sevindi. You could also write:

Hediye kartını … bıraktım. Çok sevindi.
Hediye kartını … bıraktım ve çok sevindi. Semicolons are also acceptable.

Who is the subject of sevindi if there’s no o?

Turkish drops subject pronouns when the verb ending makes the person/number clear. sevindi is 3rd person singular, so it means “he/she got very happy,” understood from context (the person whose birthday it is). For clarity you can add:

O çok sevindi.
Buna çok sevindi. (He/she was very happy about this.)

What does sevindi convey exactly? Is it the same as “was happy”?

sevinmek means “to become/glow with happiness; to be glad about something,” often tied to a specific event or reason. çok sevindi ≈ “he/she was delighted.” Nuances versus near-synonyms:

mutlu oldu = became (truly) happy (stronger, broader)
memnun oldu = was pleased/satisfied (milder, polite)

Why is çok before sevindi? Can I put it elsewhere?

Adverbs typically precede the verb or phrase they modify. çok sevindi is the natural placement. sevindi çok sounds poetic/marked. Note that çok means “very” with adjectives/verbs, but “many/much” with nouns:

çok sevindi = was very happy
çok hediye kartı bıraktım = I left many gift cards (no accusative, because it’s indefinite plural/amount)

Could I use koydum instead of bıraktım?

Yes:

koymak = to put/place (neutral, careful placing)
bırakmak = to leave/put down/leave behind (can imply leaving it there, sometimes with a hint of not taking it back) Both work here; choose based on nuance. Masaya koydum is very common for “I put it on the table.”

Can I change the word order?

Yes; Turkish is flexible, with the verb usually last. Moving elements changes emphasis:

Doğum gününde hediye kartını masaya bıraktım, çok sevindi. (emphasis on time)
Masaya hediye kartını bıraktım, çok sevindi. (emphasis on place)
Hediye kartını masaya doğum gününde bıraktım, çok sevindi. (emphasis on the fact it was on the birthday)

What are the buffer letters y and n doing here?

They prevent vowel clashes when adding a vowel-initial suffix:

masa + (Dat) -a → masaya (buffer y)
kartı + (Acc) -ı → kartını (buffer n, because the base already has a possessive/compound suffix)
günü + (Loc) -de → gününde (buffer n) General idea:
Use y after a bare vowel-final noun (masa → masaya).
Use n after a vowel-final possessed/compound form (kartı/günü → kartını/gününde).

Do I need to add something like buna with sevinmek?

You don’t have to, but you can. sevinmek commonly takes a dative complement for the cause:

Hediye kartına çok sevindi. = He/she was very happy about the gift card. In your sentence, the cause is obvious from the previous clause, so çok sevindi alone is natural.