Кроме чая, я хочу кофе.

Here кроме means besides, apart from, or in addition to.

So Кроме чая, я хочу кофе means something like:

• Besides tea, I want coffee.
• Apart from tea, I want coffee too.

It does not normally mean instead of tea.
If you wanted instead of tea, Russian would usually use вместо:

• Вместо чая я хочу кофе. = Instead of tea, I want coffee.

Because кроме is a preposition that requires the genitive case.

So:

• чай = nominative dictionary form
• чая = genitive singular

This is a very common pattern:

• кроме чая = besides tea
• кроме кофе = besides coffee
• кроме меня = besides me

So the learner-friendly rule is:

• кроме + genitive

Yes, in normal modern Russian, кроме is used with the genitive case.

Examples:

• кроме брата = besides my brother
• кроме книги = besides the book
• кроме детей = besides the children

So when you see кроме, you should expect the next word to be in the genitive.

Because хотеть normally takes a direct object, and here that object is кофе.

You might expect an accusative ending, but кофе is one of those Russian nouns that is usually indeclinable, meaning its form does not change across cases in ordinary usage.

So:

• nominative: кофе
• accusative: кофе

That is why you see:

• Я хочу кофе. = I want coffee.

Compare that with a noun that does change:

• Я хочу чай. = I want tea.
• Я хочу воду. = I want water.

Traditionally, кофе is treated as masculine in standard Russian, even though it ends in -е.

For this sentence, though, gender does not really affect the form, because кофе is indeclinable here.

You mainly notice its gender in agreement:

• горячий кофе = hot coffee
• вкусный кофе = tasty coffee

Not:

• горячее кофе in standard usage

So in я хочу кофе, the important point is not gender but the fact that кофе keeps the same form.

The phrase кроме чая is set off with a comma because it is an added phrase at the beginning of the sentence, expressing addition/exclusion: besides tea.

So the structure is:

• Кроме чая, = apart from tea / besides tea
• я хочу кофе. = I want coffee.

In sentences like this, the comma is standard and helps show that кроме чая is a separate introductory-like phrase in meaning.

Yes. Russian word order is flexible, and the basic meaning stays the same.

Possible versions include:

• Кроме чая, я хочу кофе.
• Я хочу кофе, кроме чая. — less natural in many contexts
• Я, кроме чая, хочу кофе. — possible, but more marked
• Кофе я хочу, кроме чая. — very marked, depends on context

The most natural neutral version here is the one you were given:

• Кроме чая, я хочу кофе.

It puts besides tea first, which matches the English idea Besides tea...

Often yes, if the subject is clear from context.

Russian frequently drops personal pronouns when they are not needed.

So both are possible:

• Кроме чая, я хочу кофе.
• Кроме чая, хочу кофе.

The version with я sounds a little more explicit.
The version without я can sound more natural in casual conversation when it is already obvious who is speaking.

Yes, very often that is the idea.

Even though Russian does not use a separate word for too here, кроме already creates that meaning of addition:

• Besides tea, I want coffee.
• In natural English: I want coffee too, not just tea.

So the sentence suggests that tea is also involved in some way, and coffee is an additional thing the speaker wants.

Use вместо, not кроме:

• Вместо чая я хочу кофе.

This is an important contrast:

• Кроме чая, я хочу кофе. = Besides tea, I want coffee.
• Вместо чая я хочу кофе. = Instead of tea, I want coffee.

So:

• кроме = addition
• вместо = substitution

The stress is:

• Кро́ме ча́я, я хочу́ ко́фе.

A few pronunciation notes:

• кроме = KRO-me
• чая = CHA-ya
• хочу = stress on the last syllable, kha-CHOO
• кофе = KO-fye

Also remember that unstressed о in Russian is often pronounced closer to a than a clear English o, so actual speech may sound softer than the spelling suggests.