Learn Hardcore Turkish: Arkadaşım gelince seviniyorum.

sevinmek: to feel glad/pleased (often for a specific trigger). Natural here.
mutlu olmak: to become/be happy (broader, a bit more formal/neutral): Arkadaşım gelince mutlu oluyorum .
sevindirmek: to make someone happy ( <a class="border-b border-dotted border-blue-400 dark:border-blue-500 cursor-pointer text-inherit no-underline" href="/grammar/turkish/verbs/voice-causative-dir">causative ): Arkadaşımın gelmesi beni sevindiriyor = My friend’s coming makes me happy.

It forms a time clause meaning when/whenever. So gelince = when (someone) comes. It doesn’t itself mark tense; the main clause verb (seviniyorum) sets the time frame.

Why is it gelince and not geldiğinde?

Both are correct:

Arkadaşım gelince seviniyorum is shorter and very common.
Arkadaşım geldiğinde seviniyorum uses the -dik + possessive + -de structure, which can mark the subject on the verb (e.g., geldiğimde = when I come). It can sound a bit more explicit/formal. Meaning is essentially the same here.

How does vowel harmony decide the form of -ince? What are the other shapes?

The suffix has 4-way harmony and a buffer -y- after vowels:

After front unrounded (e, i): -ince → gelince
After front rounded (ö, ü): -ünce → gülünce
After back unrounded (a, ı): -ınca → kalayınca
After back rounded (o, u): -unca → konuşunca
If the verb stem ends in a vowel, use -y-: başla → başlayınca

How is -ince different from -ken?