Σε περίπτωση που δεν προλάβεις το τρένο, θα σε πάρω τηλέφωνο.

Σε περίπτωση που means in case (that) and is a bit more formal/explicit than αν (if).

• αν δεν προλάβεις το τρένο, ... = plain if
• σε περίπτωση που δεν προλάβεις το τρένο, ... = in the event that / in case (emphasizes the contingency)

The preposition σε normally takes the accusative in Modern Greek. So περίπτωση is accusative here: σε περίπτωση. It’s a fixed expression meaning in case.

Here που introduces a clause after σε περίπτωση, roughly like that in in case that.... In everyday Greek you’ll almost always hear σε περίπτωση που + clause as a set pattern.

Because this is a conditional “future possibility” situation. Greek typically uses:

• να + subjunctive or (implied subjunctive) forms after conditional setups like αν / σε περίπτωση που when referring to something not yet known to happen.

προλάβεις is the aorist subjunctive (perfective aspect) of προλαβαίνω. It focuses on the single completed outcome: whether you manage to catch it (or not).

You can think of it that way: the verb form is the same type you’d see after να (subjunctive). After αν / σε περίπτωση που, Greek often uses the subjunctive form without explicitly adding να.
You’ll also sometimes hear σε περίπτωση να..., but σε περίπτωση που + subjunctive is very common and natural.

With τρένο (and other departures), προλαβαίνω means to make it in time / manage to catch.
So δεν προλάβεις το τρένο = you don’t make it in time to catch the train.

Greek negation δεν (not) goes directly before the verb: δεν προλάβεις.
(With subjunctive introduced by να, the negation would be μη(ν), but here you have δεν because the clause is structured without να.)

θα marks the future (and sometimes “would” in other contexts). Here it’s a straightforward future: I will call you.

In Greek, short object pronouns (clitics) normally come before the main verb:

• θα σε πάρω = I’ll call you
  Not θα πάρω σε (that would be unnatural in most normal speech).

σε can be both you (singular) and you (plural) in this unstressed form. Context decides.
If you want to be explicit in writing, you might add:

• singular: θα σε πάρω (εσένα) τηλέφωνο
• plural: θα σας πάρω τηλέφωνο (σας = you-plural / polite you)

It’s an idiom: παίρνω (κάποιον) τηλέφωνο = to call (someone) on the phone.
Other common equivalents are:

• θα σου τηλεφωνήσω = I’ll phone you (more “verb-like”)
• θα σε καλέσω = I’ll call you

It separates the introductory conditional clause from the main clause, just like in English:
In case ..., I’ll ...
Greek punctuation often uses a comma in this structure.

Yes. Greek allows flexibility for emphasis:

• Σε περίπτωση που δεν προλάβεις το τρένο, θα σε πάρω τηλέφωνο. (neutral, condition first)
• Θα σε πάρω τηλέφωνο σε περίπτωση που δεν προλάβεις το τρένο. (puts the promise first)
  Both are natural; the first is slightly more common in careful speech/writing.