Learn Hardcore Greek: Παρακαλώ, δώστε μου ένα νερό και ένα λεπτό.

Because δώστε is the 2nd person plural imperative, used for polite/formal address or when speaking to more than one person. δώσε is singular/informal, for a friend or someone you’re on first-name terms with.

Formal to a waiter: Δώστε μου ένα νερό…
Informal to a friend: Δώσε μου ένα νερό…

What exactly is the form δώστε (tense/aspect), and why use it here?

Δώστε is the aorist imperative of δίνω (to give). Greek imperatives contrast aspect:

Aorist imperative (here): one-off, complete action — perfect for a single request: Δώστε μου…
Present imperative: ongoing/repeated action — e.g., Δίνετε μου… would suggest “keep giving me…,” which is not what you want when ordering.

Where does the pronoun μου go? Could I say μου δώστε?

With a positive imperative, weak pronouns go after the verb: δώστε μου. In other environments they typically go before:

Positive imperative: Δώστε μου ένα νερό.
Non-imperative: Μου δίνετε ένα νερό;
Negative imperative-like (subjunctive): Μην μου δώσετε νερό.

Do I need μου at all? Can I just say “give a water”?

You can drop it in ordering contexts. The most natural, concise way to order is simply: Ένα νερό, παρακαλώ.
Adding μου (to me) is fine and a bit more explicit/personal: Δώστε μου ένα νερό.

Why ένα νερό if “water” is uncountable in English?

In Greek, you can count portions/servings. Ένα νερό essentially means “one water (serving),” typically a bottle or glass. If you want to be explicit:

Ένα μπουκάλι νερό (a bottle of water)
Ένα ποτήρι νερό (a glass of water)

Why repeat ένα before both nouns? Could I say ένα νερό και λεπτό?

Repeat it. Greek normally repeats the article/numeral with coordinated nouns: ένα νερό και ένα λεπτό. Omitting it before the second noun (ένα νερό και λεπτό) sounds off. You can also use the clitic form of “and” before a vowel: …κι ένα λεπτό.

What does ένα λεπτό mean here? Is it “thin”?

Here (το) λεπτό is the noun “minute,” used idiomatically as “a moment.” It is not the adjective λεπτός (“thin”). Note:

Time: ένα λεπτό (one minute), δύο λεπτά (two minutes)
Money: λεπτό also means a cent (of the euro) Common alternatives: Μισό λεπτό! (half a sec), Μια στιγμή! (one moment)

Is it idiomatic to ask for a minute in the same breath as ordering?

Grammatically yes, stylistically a bit unusual. More natural is to split it:

Ένα νερό, παρακαλώ. Και δώστε μου ένα λεπτό. Or simply say to someone waiting on you:
Ένα λεπτό, παρακαλώ.

How do I pronounce the sentence?

Παρακαλώ: [pa-ra-ka-LO] (stress on the last syllable)
δώστε: [THO-ste] (δ = voiced “th” as in “this”; stress on the first syllable)
μου: [moo]
ένα: [E-na] (stress on the first syllable)
νερό: [ne-RO]
λεπτό: [lep-TO] Note: ο and ω sound the same in Modern Greek; the accent mark shows stress, not vowel quality.

Can και become κι here?

Yes. και becomes κι before a vowel sound for smoother flow. Both are correct:

…και ένα λεπτό
…κι ένα λεπτό

Why ένα and not έναν?

Because both νερό and λεπτό are neuter nouns. The neuter form is ένα (nominative/accusative). Έναν is masculine accusative, used with masculine nouns:

έναν καφέ (a coffee, masc.)
ένα νερό, ένα λεπτό (neuter)

Other polite ways to ask?

Ένα νερό, παρακαλώ.
Θα ήθελα ένα νερό.
Μπορείτε να μου δώσετε ένα νερό;
Σας παρακαλώ, ένα νερό.
Even more tentative: Μήπως μπορείτε να μου δώσετε ένα νερό;

Is the comma after Παρακαλώ required?

Not required, but common because Παρακαλώ functions as an interjection. All of these are fine:

Παρακαλώ, δώστε μου…
Δώστε μου, παρακαλώ, …
Δώστε μου …, παρακαλώ.

Does Παρακαλώ also mean “You’re welcome” or “Hello?” on the phone?

Yes. Παρακαλώ is versatile:

“Please” in requests
“You’re welcome” after thanks
“Hello?/Yes?” when answering the phone

How do I make it plural or otherwise vary quantity?

Two waters: Δώστε μου δύο νερά. (or more explicit: δύο μπουκάλια νερό)
Two minutes: Δύο λεπτά, παρακαλώ.
Very common softener: Μισό λεπτό, παρακαλώ.

How do I say it informally to a friend?

Δώσε μου ένα νερό κι ένα λεπτό. Even more casual ordering style (no verb): Ένα νερό, ρε. Μισό λεπτό.

How do I say “Don’t give me water; give me a minute” (to see the negative pattern)?

Μη(ν) μου δώσετε νερό· δώστε μου ένα λεπτό. Note the pronoun placement with the negative: μη(ν) + μου + δώσετε.