Learn Hardcore Croatian: Narudžba još nije stigla, pa ću ih zvati odmah. - The order still hasn’t arrived, so I will call them immediately.

Stigla is the past participle (l-participle) of stići (to arrive), and it agrees in gender and number with the subject narudžba (order), which is feminine singular.
So you get narudžba … stigla.
If the subject were masculine, you’d see stigao; neuter stiglo; plural stigli/stigle/stigla.

What tense is još nije stigla? Is it “present perfect”?

It’s the standard Croatian past tense built with biti (to be) + l-participle: nije stigla = hasn’t arrived / didn’t arrive (depending on context).
Croatian doesn’t label it “present perfect” the way English does; it’s simply the normal past tense form used for many meanings English splits across didn’t vs hasn’t.

Why is the auxiliary nije in the middle: još nije stigla?

Croatian commonly places the clitic auxiliary (je/nisam/nije, etc.) early in the clause, often in “second position” (after the first stressed element).
Here još is first, so the clitic nije follows it: još + nije + stigla.
You may also see: Narudžba nije još stigla, but još nije stigla is very natural.

What exactly does još mean here?

Još here means yet / still. With a negative verb it often corresponds to English not … yet:

još nije stigla = hasn’t arrived yet.

What does pa mean—does it always mean “and”?

Here pa is a connector meaning so / therefore / in that case.
It often introduces a consequence: X, pa Y = X, so Y.
It can also mean “and then” in some contexts, but in this sentence it’s clearly causal: the order hasn’t arrived, so I’ll call.

How is the future formed in pa ću ih zvati?

That’s Future I: the clitic form of htjeti (will) + infinitive.

ću (I will) + zvati (to call) = ću zvati (I will call).
Other persons: ćeš, će, ćemo, ćete, će.

Why is it ću ih zvati and not zvat ću ih? Which is correct?

Both are correct and common.

ću ih zvati odmah (auxiliary first)
zvat ću ih odmah (infinitive first; zvat is a shortened colloquial infinitive of zvati)
Clitic placement rules still apply: the clitic cluster (ću ih) typically comes right after the first stressed element in the clause.

What is ih, and why is it before zvati?

Ih is the unstressed (clitic) accusative form meaning them.
Clitic pronouns in Croatian usually come early in the clause, often right after the first stressed element, and before the main verb/infinitive:

ću ih zvati = I’ll call them.

Who does ih refer to? Why not a pronoun for narudžba?

Narudžba is the order, but you don’t usually “call an order”; you call the people responsible for it (a shop, courier service, customer support, etc.). That’s why ih = them (the staff/people/company).
If you meant calling one person, you’d use ga (him) / je (her) / vas (you-pl./formal), etc.

Is zvati the same as nazvati or pozvati?

Related, but not identical:

zvati = to call (general; includes phone calls and “call someone by a name”)
nazvati (nekoga) often emphasizes “to phone/call” and is very common for phone calls
pozvati (nekoga) = to invite / to call on (summon)
In this sentence, zvati works fine for “phone them,” but nazvati could also fit: pa ću ih nazvati odmah.

What does odmah mean, and where can it go in the sentence?

Odmah means immediately / right away.
It’s fairly flexible in position:

pa ću ih zvati odmah
pa ću ih odmah zvati
Both are natural; placement can slightly affect emphasis.

Any pronunciation/letter issues I should notice in Narudžba and još?

Two key letters:

ž (as in narudžba) sounds like the s in measure (/ʒ/).
š (as in još) sounds like sh (/ʃ/).
Also, Croatian r can act like a vowel in some words, but not in these.