Learn Turkish: Adam kendi kendine gülümsedi.

The phrase kendi kendine acts as an adverb translating to "to himself" or "by himself." The first kendi is the base word, while the second takes the dative case ending -e (meaning "to"). Literally, it means "himself to himself."

Could we just say 'Adam kendine gülümsedi' without doubling the word?

Yes, but it changes the feel of the sentence. Using just kendine simply means "to himself." The reduplicated adverb kendi kendine adds a strong emphasis on the solitary, internal nature of the action—he was completely lost in his own thoughts, smiling all by himself.

If the subject changes to "I" or "we", does the first kendi change too?

No, only the second kendi changes to match the subject. For "I smiled to myself," you would say Kendi kendime gülümsedim (adding the first-person -im plus the dative -e to the second word). For "we," it would be kendi kendimize. The first word always stays as the bare kendi.

What is the difference between gülümsedi and güldü in this context?

Gülümsedi (from gülümsemek) strictly means "smiled." If you used güldü (from gülmek), the sentence would mean he "laughed" to himself.