Learn Hardcore German: Kannst du kurz auf mein Handgepäck achten, während ich meinen Sitzplatz finde? - Can you keep an eye on my carry-on for a moment while I find my seat?

Starting with Kannst du …? is the standard way to form a yes/no question in German: the conjugated verb (kannst) comes first, then the subject (du).
Du kannst …? can also be a question in speech (with rising intonation), but it’s more informal/colloquial and can sound a bit like confirmation or surprise.

What exactly does kurz mean here, and where can it go in the sentence?

kurz here means for a moment / briefly (i.e., not for long). In your sentence it modifies the request auf mein Handgepäck achten.
Common placements:

Kannst du kurz auf mein Handgepäck achten …? (very natural)
Kannst du auf mein Handgepäck kurz achten …? (possible, but less smooth)
Kannst du kurz darauf achten, …? (also common if you use darauf)

What does auf … achten mean literally, and what’s the best English sense?

Literally, achten is to pay attention, and auf … achten is to pay attention to ….
In everyday situations like luggage, it usually means to keep an eye on / to watch / to look after:

auf mein Handgepäck achten = watch my carry-on (for a moment)

Why is it auf mein Handgepäck (accusative) and not dative?

With achten auf, the preposition auf takes the accusative because it’s not expressing location here; it’s part of a fixed verb + preposition combination meaning pay attention to.
So you get:

auf + Akkusativ → auf mein Handgepäck

What gender is Handgepäck, and why is it mein (not meine or meinen)?

Handgepäck is neuter: das Handgepäck.
In the accusative singular, mein stays mein for neuter nouns:

nominative/accusative neuter: mein Handgepäck
(Compare: meine Tasche (feminine), meinen Koffer (masculine accusative).)

Is Handgepäck the same as Handtasche or Koffer?

Not exactly:

Handgepäck = carry-on luggage (the general category: backpack, small suitcase, tote, etc., as opposed to checked luggage)
Handtasche = handbag/purse
Koffer = suitcase
So Handgepäck is a broader, travel-context word.

Why is there a comma before während, and why does the verb move to the end: … während ich meinen Sitzplatz finde?

während introduces a subordinate clause, and German normally separates subordinate clauses with a comma.
In subordinate clauses, the conjugated verb goes to the end:

main clause: Kannst du … achten?
subordinate clause: während ich … finde (verb finde at the end)

Does während require a specific case, and is it being used as a preposition or a conjunction here?

Here während is a conjunction meaning while, so it introduces a clause (während ich … finde) and does not assign case.
As a preposition, während can mean during and traditionally takes the genitive (often genitive in formal German, sometimes dative in everyday speech):

preposition: während des Flugs = during the flight

Why is it meinen Sitzplatz (accusative), and what nuance does Sitzplatz have?

finden takes a direct object, so meinen Sitzplatz is accusative. Sitzplatz is masculine: der Sitzplatz, so accusative is meinen.
Nuance:

Sitzplatz = your assigned seat (typical on planes/trains with reservations)
Platz can be more general (spot/seat/place), but Sitzplatz is clearer in travel contexts.

How would this sound in a more polite/formal way?

You can switch to Sie and/or add bitte:

Könnten Sie bitte kurz auf mein Handgepäck achten, während ich meinen Sitzplatz finde?
Using könnten (subjunctive) is softer/more polite than kannst.