Learn Hardcore Turkish: Ne kadar şaşkın olursam olayım, önce sakinleşmek istiyorum. - However surprised I am, I want to calm down first.

It expresses a concessive idea: "no matter how ... I am/may be." Here, Ne kadar şaşkın olursam olayım means "No matter how bewildered I may be." You concede the state (being surprised to any degree) but go on to state your main point.

Why are both verbs "olursam" and "olayım" used together?

This is a fixed, emphatic concessive pattern: conditional + optative of the verb ol-.

olursam = if I am/become
olayım = may/let me be (optative) Together they mean "however I may be," not two separate actions. It’s similar in force to "even if I’m X (to any degree)."

Can I replace it with "Ne kadar şaşkın olsam da"?

Yes. Ne kadar şaşkın olsam da, ... is a very common and slightly simpler alternative with the same meaning: "Even if/however surprised I am, ..." The "olursam olayım" version is a bit more emphatic/formulaic.

What about "Her ne kadar ... olsa da"? Is that the same?

Close, but slightly different in feel. Her ne kadar şaşkın olsam da, ... means "Although I’m very/really surprised, ..." It’s often a bit more formal. Ne kadar ... tends to mean "no matter how ...", highlighting degree; her ne kadar ... olsa da is more like "although."

Why do we need "ol-" with an adjective like "şaşkın"?

Adjectives and nouns in Turkish need a verb to form a predicate in subordinate clauses. ol- ("to be/become") supplies that verb: şaşkın ol- = "to be surprised/bewildered." So the verbal forms (olursam, olayım) carry the adjective.

Can you break down "olursam" and "olayım" morphologically?

olursam = ol- (be/become) + -ur (aorist) + -sa (conditional) + -m (1sg) → "if I am/become"
olayım = ol- (be/become) + -a (optative) + -yım (1sg, with buffer -y-) → "may I be" (in this construction: "I may be, however that is")

Is "olayım" expressing a wish ("let me be") here?

Literally, it is the 1st-person optative, but in this frozen pattern (olursam olayım) it does not express a wish. It simply completes the concessive idea "however I may be." Think of it as an idiomatic formula.

Could I just say "Ne kadar şaşkın olursam, ..." without "olayım" or "da"?

No. That would leave the conditional clause hanging. You need either:

the full formula: Ne kadar şaşkın olursam olayım, ..., or
the "even though" version: Ne kadar şaşkın olsam da, ...

What nuance does "şaşkın" have compared to "şaşırmış" or "şok olmuş"?

şaşkın: bewildered, flustered, at a loss (an ongoing state)
şaşırmış: surprised/astonished (result state of having been surprised)
şok olmuş: shocked (stronger, often sudden/acute) Here, şaşkın suggests being rattled and not fully composed.

How do I pronounce the tricky letters here, especially in "şaşkın" and "sakinleşmek"?

ş = "sh" (English "shoe")
ı (dotless i in şaşkın) = a close back unrounded vowel, like the second vowel in "roses" for many speakers; not like English "ee" or "ih"
i (dotted i in sakinleşmek) = "ee" (English "machine") So: şaşkın ≈ "shush-kun" (with that special ı), sakinleşmek ≈ "sah-keen-lesh-mek."

What does "sakinleşmek" mean exactly, and how is it different from "sakin olmak" or "sakinleştirmek"?

sakinleşmek: to calm down, become calm (inchoative/reflexive; change of state)
sakin olmak: to be calm (a state)
sakinleştirmek: to calm someone/something (causative) In the sentence, sakinleşmek istiyorum = "I want to calm down."

Why is "istiyorum" (present continuous) used for "I want"? Could I say "isterim" or "sakinleşeyim"?

istiyorum is the default, natural way to express a present desire/intention: "I want (now)."
isterim can feel habitual or tentative/polite ("I would like") depending on context.
Önce sakinleşeyim (optative) means "Let me calm down first"—a self-directive suggestion/decision, a bit stronger than just "I want to."

What does "önce" modify here, and can I move it?

önce means "first/first of all." It modifies the whole intention: "First, I want to calm down." You can move it for emphasis:

Önce sakinleşmek istiyorum. (neutral, common)
Sakinleşmek istiyorum önce. (puts light emphasis on "before anything else") Keep the comma after the concessive clause: Ne kadar …, önce …

Is there a generic, person-neutral version of this pattern?

Yes: Ne kadar X olursa olsun, ... (3rd person). Example: Ne kadar zor olursa olsun, vazgeçme. = "No matter how hard it is, don’t give up." Your sentence uses the 1st-person version: olursam olayım.

Can I use this "ne kadar ... -sa ... -yım" pattern with verbs too?

Yes, with verbs you duplicate the same verb:

Ne kadar uğraşırsam uğraşayım, pes etmeyeceğim. = "No matter how much I try, I won’t give up." With adjectives/nouns, you use ol-: Ne kadar şaşkın olursam olayım...

Why do we see -ur in "olursam" but not in "olsam"?

Turkish has two ways to form such conditionals:

with the aorist before -sa: ol-ur-sa-m → olursam
directly with -sa on the stem: ol-sa-m → olsam In the "ne kadar ... olursa/olursam ... olsun/olayım" concessive formula, the aorist version is conventional. The simpler alternative uses the bare conditional plus da: olsam da.