Learn Hardcore Ukrainian: Мені хочеться морозива після роботи.

Мені хочеться is an impersonal construction that literally means “It is wanting to me” or more naturally “I feel like (it)”.
It sounds softer and more spontaneous than Я хочу, which is a direct “I want” and can feel stronger or more demanding.
So:

Мені хочеться морозива ≈ “I feel like (having) some ice cream.”
Я хочу морозива ≈ “I want (some) ice cream.”

Both are correct; the sentence just chooses the softer, more “mood/feeling”-based version.

What exactly is хочеться grammatically? Is it a reflexive verb?

Yes. Хочеться is the 3rd person singular present of the reflexive verb хотітися.

Base verb: хотіти – “to want”
Reflexive form: хотітися – “to feel like (wanting something)”
3rd person singular reflexive: хочеться

It’s used in impersonal sentences, where there is no grammatical subject like “I/you/he”. Instead, the experiencer is in the dative (мені, тобі, йому, etc.), and the verb stays in 3rd person singular:
Мені хочеться…, Тобі хочеться…, Йому хочеться…

What case is Мені, and why is it used here instead of Я?

Мені is the dative case of я (“I”).

In Ukrainian, feelings, states, and spontaneous desires are very often expressed with the dative:

Мені холодно. – “I am cold.” (literally “To me it is cold.”)
Їй сумно. – “She is sad.” (literally “To her it is sad.”)
Нам хочеться спати. – “We feel like sleeping.”

So Мені хочеться… fits that pattern: “To me it is wanting,” i.e. “I feel like it.”
Using nominative Я with a personal verb (Я хочу…) is also possible, but that’s a different, more direct structure.

Why is it морозива and not морозиво here?

Морозива is the genitive singular of морозиво (“ice cream”).
In Ukrainian, when you want “some (amount of) a thing”, especially a mass or uncountable noun, the object is very often in the genitive:

Хочу води. – “I want (some) water.”
Хочу хліба. – “I want (some) bread.”
Мені хочеться морозива. – “I feel like (some) ice cream.”

So морозива suggests an indefinite amount – you’re not pointing to one specific “ice cream item,” just craving ice cream in general.

Can I say Мені хочеться морозиво instead? Is it wrong?

It’s not wrong, but it can sound a bit different.

Roughly:

Мені хочеться морозива – neutral, very natural, “I feel like (some) ice cream.”
Мені хочеться морозиво – can sound a bit more like wanting a specific ice cream as an item (like “an ice cream”), and is less common in this particular pattern.

You will frequently hear Хочу морозиво! from children, for example, when they want a specific ice cream treat.
In your sentence, though, морозива is the most idiomatic choice.

Why is it після роботи and not після робота?

Because the preposition після (“after”) always takes the genitive case.

робота – nominative (dictionary form)
роботи – genitive singular

So:

після роботи – “after work”
This is the same pattern you’ll see with після in other phrases:
після уроку (after the lesson), після фільму (after the movie), після обіду (after lunch).

Can I change the word order, for example Після роботи мені хочеться морозива?

Yes, Ukrainian word order is quite flexible, and all of these are grammatical:

Мені хочеться морозива після роботи.
Після роботи мені хочеться морозива.
Мені після роботи хочеться морозива.

They all mean the same basic thing.
Moving після роботи to the beginning (Після роботи…) puts a bit more emphasis on the time (“After work, I feel like some ice cream”), but it’s a nuance of focus, not a change of meaning.

How would I say this for other people, not just “me”?

You keep хочеться the same and change only the dative pronoun:

Мені хочеться морозива після роботи. – I feel like ice cream…
Тобі хочеться морозива після роботи. – You (sg.) feel like ice cream…
Йому хочеться морозива після роботи. – He feels like ice cream…
Їй хочеться морозива після роботи. – She feels like ice cream…
Нам хочеться морозива після роботи. – We feel like ice cream…
Вам хочеться морозива після роботи. – You (pl./formal) feel like ice cream…
Їм хочеться морозива після роботи. – They feel like ice cream…

Notice that хочеться does not change with person; only the dative pronoun changes.

How would I negate this sentence?

You simply add не before хочеться:

Мені не хочеться морозива після роботи. – “I don’t feel like ice cream after work.”

The pattern is:

Мені не хочеться…
Тобі не хочеться…
Їм не хочеться… etc.

The не goes directly in front of the verb.

If I am a woman, do I need to change anything in this sentence?

No, nothing changes.

The sentence uses:

Мені – dative pronoun (no gender distinction)
хочеться – impersonal 3rd person singular
морозива, роботи – nouns (whose forms don’t depend on your gender)

So Мені хочеться морозива після роботи works the same for a man or a woman.

How could I ask someone “Do you feel like ice cream after work?” in Ukrainian?

You can use the same impersonal pattern with тобі:

Тобі хочеться морозива після роботи? – “Do you feel like (having) ice cream after work?”

More neutral / direct with хотіти:

Ти хочеш морозива після роботи? – “Do you want ice cream after work?”

Both are correct; Тобі хочеться…? sounds a bit more about mood/craving, Ти хочеш…? is a straightforward “Do you want…?”

Where is the stress in the words Мені хочеться морозива?

Standard stress is:

МенІ – me-NEE
ХОчеться – KHO-che-tsia (stress on the first syllable)
морОзива – mo-RO-zy-va (stress on the second syllable)

So you get:
МенІ ХОчеться морОзива пІсля робОти.