Learn Hardcore Turkish: Ben kardeşime sürpriz yapmak istiyorum.

Because sürpriz yapmak treats the person who receives the surprise as a recipient/goal, which is marked with the dative. Think of it as do a surprise to someone:

Kime sürpriz yapıyorsun? — Kardeşime.

Could I say kardeşimi sürpriz yapmak?

No. kardeşimi (accusative) would mark a direct object, but with sürpriz yapmak the person is a recipient, so use dative: kardeşime sürpriz yapmak. If you want a verb that takes the person as a direct object, use şaşırtmak: Kardeşimi şaşırtmak istiyorum (I want to surprise my sibling).

Is Ben necessary here?

No. The verb ending already shows the subject. Kardeşime sürpriz yapmak istiyorum is perfectly natural. Including Ben adds emphasis or contrast (I, as opposed to someone else).

Why is it istiyorum (present continuous) to say “I want”?

Turkish commonly uses the -yor (present continuous) for current desires and intentions. İstiyorum = I want (now/definitely). İsterim (aorist) sounds general, hypothetical, or polite/softened (I’d like / I would like).

What does yapmak do here? Could I just say sürpriz istiyorum?

Yapmak is a light verb meaning do/make; with sürpriz it forms the idiomatic verb do a surprise (to someone): sürpriz yapmak.
Sürpriz istiyorum means I want a surprise (for myself), not I want to surprise someone.

Do I need bir? As in Bir sürpriz yapmak istiyorum?

Both are fine:

Sürpriz yapmak istiyorum: natural, generic “do a surprise.”
Bir sürpriz yapmak istiyorum: emphasizes a single, specific instance “I want to do a (one) surprise.”

Why not sürpriz etmek or a single verb for “to surprise”?

The standard collocation is sürpriz yapmak. You’ll also hear sürpriz etmek, but it’s much less common/natural in this meaning. The single-verb option is şaşırtmak (to surprise someone): Kardeşimi şaşırtmak istiyorum.

What exactly does kardeş mean—brother or sister?

Kardeş is gender-neutral (sibling). To be specific:

kız kardeş = sister (younger or unspecified age)
erkek kardeş = brother (younger or unspecified age)
abla = older sister
ağabey/abi = older brother

With your sentence’s case/possessive:

kız kardeşime, erkek kardeşime, ablama, abime.

How flexible is the word order?

Turkish likes the finite verb at the end, but the other parts can move for emphasis:

Kardeşime sürpriz yapmak istiyorum (neutral)
Sürpriz yapmak istiyorum kardeşime (emphasizes the recipient)
Ben kardeşime sürpriz yapmak istiyorum (emphasizes I)

Any pronunciation tips for sürpriz and istiyorum?

ü is a front rounded vowel (like German ü or French u in tu).
r is tapped/flapped.
sürpriz: roughly “sür-priz,” stress often on the last syllable.
istiyorum: “is-tee-yo-rum,” the -yor is pronounced as “yor.”

Can I say kardeşim için sürpriz yapmak istiyorum?

Yes. … için means “for (the sake/benefit of).”

kardeşime sürpriz yapmak = do a surprise to my sibling (recipient).
kardeşim için sürpriz yapmak = do a surprise for my sibling (for their sake).
Both are acceptable; the dative is the more idiomatic frame with sürpriz yapmak.

How do I know whether to use -e or -a for the dative?

Vowel harmony:

If the last vowel of the word is front (e, i, ö, ü) → -e
If it’s back (a, ı, o, u) → -a
In kardeşim, the last vowel is i (front), so kardeşime.

What if I want to say “I will surprise my sibling” instead?

With the collocation: Kardeşime sürpriz yapacağım (I will make a surprise for/to my sibling).
With the single verb: Kardeşimi şaşırtacağım (I will surprise my sibling).

Why is sürpriz not accusative (sürprizi) here?

Because it’s an indefinite direct object; in Turkish, indefinite objects are unmarked (no accusative). Sürprizi would mean the surprise (specific/known) or his/her surprise, which changes the meaning.