Learn Hardcore Turkish: Ben soyadımı yavaşça heceliyorum.

soyadım = “my surname” (bare form; usually subject/predicate)
soyadımı = “my surname” as a definite object
“Soyadım heceliyorum” is ungrammatical; you need the -ı.

How is "soyadımı" built?

soyad-ım-ı

soyad = family name/surname
-ım = my (1st person singular possessive; chosen by vowel harmony)
-ı = accusative (definite object; also chosen by vowel harmony)
No buffer letter is needed because the word ends in a consonant (m).

What exactly is in "heceliyorum," and why not "heceleiyorum"?

hecele- (stem) + -(I)yor (present progressive) + -um (1sg)
When a verb stem ends in a vowel, that final vowel drops before -(I)yor:

hecele- + -iyor → heceliyor (not heceleiyor)
Compare: bekle- + -iyor → bekliyor; anla- + -ıyor → anlıyor.

Why is the ending -um, not -ım, for “I”?

After -(I)yor, the 1st person singular ending is always -um due to rounding harmony triggered by the -yor.
Examples: geliyorum, yapıyorum, istiyorum, heceliyorum.

What’s the difference between "heceliyorum" and "hecelerim"?

heceliyorum = I am spelling (right now / currently / in progress)
hecelerim = I spell (habitually/generally; also potential: “I can/usually do”)
Your sentence needs the progressive because it’s an ongoing action.

Does "hecelemek" mean “to spell” letter by letter or “to syllabify”?

Both, depending on context. In everyday use, hecelemek is used for “to spell out.”
For explicit letter-by-letter, people also say:

harf harf söylemek (to say it letter by letter)
kodlamak (to spell using the telephone alphabet)
“Pronounce” is different: telaffuz etmek.

Where can "yavaşça" go? Is word order fixed?

Turkish word order is flexible. Common options:

Ben soyadımı yavaşça heceliyorum. (neutral)
Ben yavaşça soyadımı heceliyorum. (slight focus on the manner)
Yavaşça soyadımı heceliyorum. (fronted adverb for emphasis on “slowly”)
All are acceptable; object-before-verb is the unmarked default.

Why "yavaşça" instead of just "yavaş"?

Both can work as adverbs.

yavaş: very common, plain “slow(ly)”
yavaşça: mannered “slowly/gently/softly,” a touch more formal or delicate
Related:
yavaş yavaş = “gradually/little by little”
ağır ağır = “very slowly, at a measured pace”

Why is it spelled "yavaşça" (with ç), not "yavaşca"?

The adverbial suffix is -cA. After a voiceless consonant (like ş), the c devoices to ç.
Vowel harmony also applies (a vs e), so yavaş + -ca → yavaşça.

Can I put the adverb at the end: "Soyadımı heceliyorum yavaşça"?

You can, especially in speech, to add an afterthought or extra emphasis on the manner. The neutral place for manner adverbs is before the verb phrase, so pre-verbal is more typical in careful style.

Can the object come after the verb: "Ben heceliyorum soyadımı"?

It’s possible but marked. Post-verbal objects signal strong focus or appear in certain conversational/emphatic contexts. Neutral Turkish prefers the object before the verb.

Is "soyadı" the same as "soyisim"? Which should I use?

They’re near-synonyms. “Soyadı” is the standard/official term; “soyisim” is common in speech.

my surname: soyadım / soyismim
my surname (object): soyadımı / soyismimi

Why isn’t there an article like “the”?

Turkish has no definite article. Definiteness of direct objects is shown by the accusative -ı/-i/-u/-ü. Indefiniteness can be shown with bir (“a/an”), but not used here because “my surname” is inherently definite.

Any pronunciation tips for this sentence?

ı (dotless i) in soyadımı is a central, unrounded vowel (like the e in “taken” for many speakers, but without rounding).
c in hece is like English j in “jam”; ç in yavaşça is like ch in “church.”
-(I)yor is stressless; the stress falls just before it: heceLIyorum.
yavaşça: pronounce ş as “sh” and ç as “ch.”

Could I use "yavaş yavaş" instead of "yavaşça"? What’s the nuance?

Yes, but it shifts the meaning to “gradually/little by little.”

yavaşça heceliyorum = I’m spelling it slowly/gently (manner)
yavaş yavaş heceliyorum = I’m spelling it little by little (stepwise, taking my time)