Hediye kartını doğum gününde masaya bıraktım, çok sevindi.

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).

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)

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

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

• bırak- + -tı + -m → bıraktım

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.

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.)

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)

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)

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.”

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)

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).

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.