Learn Hardcore Swedish: Jag vill ha en kopp kaffe utan socker.

en kopp kaffe, ett glas vatten, en flaska vin Use av only for special meanings, like material or part–whole:
en kopp av glas (a cup made of glass)
en kopp av kaffet (a cup of the coffee [that we made earlier])

Why is it en and not ett before kopp?

Because kopp is a common-gender noun (an “en-word”): en kopp.

Forms: en kopp (sing.), koppen (def. sing.), koppar (pl.), kopparna (def. pl.)

Why is there no article on kaffe? Why not kaffet?

In en kopp kaffe, kaffe is a mass noun used as a complement to the measure noun, so it stays bare (no article). kaffet means “the coffee” (specific or previously mentioned):

Kaffet är varmt. (The coffee is hot.)
But: en kopp kaffe (a cup of coffee)

Can I just say Jag vill ha kaffe or even en kaffe?

Jag vill ha kaffe is natural if quantity isn’t important.
In cafés, people often say en kaffe (treating coffee as a countable item). It’s very common in speech and ordering contexts.

What’s the difference between utan socker and inget socker?

utan socker = “without sugar” (preposition + noun), ideal after a drink: kaffe utan socker.
inget socker = “no sugar” (negative determiner + noun). Use when stating presence/absence: Kaffet innehåller inget socker. Note: socker is an “ett-word,” so it pairs with inget (not “ingen”).

Where does inte go if I want to negate this sentence?

Put inte after the finite verb vill and before the infinitive ha:

Jag vill inte ha en kopp kaffe utan socker.

Why isn’t there att before ha (why not vill att ha)?

Modal verbs like vill, kan, ska, måste, bör, får take a bare infinitive (no att):

Jag vill ha …, Jag kan dricka …

Can I move utan socker earlier in the sentence?

The natural spot is after the noun phrase it modifies:

Jag vill ha en kopp kaffe utan socker. Fronting is possible but marked: Utan socker vill jag ha en kopp kaffe (emphatic/poetic). Don’t put utan socker before kaffe inside the noun phrase.

Is Jag vill ha … polite enough when ordering? Any softer alternatives?

It’s acceptable and common, especially with tack:

Jag vill ha en kopp kaffe utan socker, tack. Softer forms:
Jag skulle vilja ha … (I would like to have …)
Kan jag få …? (Can I get …?)

How do I say “two cups of coffee without sugar”?

Jag vill ha två koppar kaffe utan socker. Note: Plural of kopp is koppar; kaffe stays unchanged as a mass noun.

Any quick pronunciation tips for this sentence?

Double consonants (pp, ff, kk) signal a short preceding vowel: kopp, kaffe, socker all have short vowels before the doubled consonants.
utan has the Swedish fronted u sound (like a rounded “ee”): long vowel in u.
Primary stress typically on the first content word syllables: kópp, káffe, sócker.

Can I use mugg instead of kopp?

Yes, en mugg is a mug (often larger, thicker). Both are fine:

en kopp kaffe (a cup of coffee)
en mugg kaffe (a mug of coffee) In cafés you might also specify size: en stor mugg kaffe.

Does svart kaffe already imply no sugar?

svart kaffe means “black coffee,” i.e., no milk/cream. It doesn’t automatically exclude sugar. To be explicit: svart kaffe utan socker.

When would I actually use av with kopp or kaffe?

Material: en kopp av glas (a cup made of glass).
Part of a specific whole: en kopp av kaffet (a cup of the coffee we have). Not for the generic “cup of coffee,” which is simply en kopp kaffe.

Is utan also used to mean “not … but (rather) …”?

Yes, utan can mean “but rather” after a negation:

Inte kaffe, utan te. (Not coffee, but tea.) In your sentence, utan is the preposition “without” in utan socker.