Kan du fylla mitt glas med vatten från kranen?

Because this is a question.

In Swedish, yes/no questions are usually made by putting the verb first:

• Du kan fylla mitt glas. = You can fill my glass.
• Kan du fylla mitt glas? = Can you fill my glass?

So kan comes before du because the speaker is asking a question.

Because fylla is the infinitive form, meaning to fill.

After a modal verb like kan = can, Swedish normally uses the infinitive without att:

• Kan du fylla ... ? = Can you fill ... ?
• Jag kan läsa. = I can read.

Compare:

• fylla = to fill
• fyller = fills / is filling
• fyll! = fill! (command)

So kan du fylla literally works like can you fill.

Because glas is an ett-word in Swedish.

Swedish nouns are divided into two grammatical genders:

• en-words
• ett-words

The possessive changes to match the gender of the noun:

• min for common gender (en words)
• mitt for neuter (ett words)
• mina for plural

Since it is:

• ett glas = a glass

you get:

• mitt glas = my glass

Not:

• min glas

It can mean both, depending on context.

• ett glas can mean a glass (drinking vessel)
• glas can also mean glass as a material

In this sentence, mitt glas clearly means my drinking glass, because it is something being filled with water.

Because Swedish uses med in this pattern, just like English uses with:

• fylla något med något = fill something with something

So:

• fylla mitt glas med vatten = fill my glass with water

Without med, the sentence would sound wrong in standard Swedish.

Because vatten is being used as an uncountable noun, like water in English.

Just as English normally says:

• with water

and not

• with a water

Swedish says:

• med vatten

and not usually

• med ett vatten

So no article is needed here.

från kranen means from the tap.

Here:

• från = from
• kranen = the tap

The basic noun is:

• en kran = a tap / faucet

The definite form is made with an ending:

• kranen = the tap

So Swedish often says the tap with the definite ending attached to the noun, instead of using a separate word like English the.

Grammatically, yes, but it would mean from a tap, which is less natural in this situation.

If you are talking about the normal tap in the kitchen or bathroom, Swedish usually uses the definite form:

• från kranen = from the tap

Using en kran sounds more like you mean some tap, any tap, a certain tap rather than the expected one.

It is usually perfectly normal and polite in everyday Swedish, especially with the right tone.

• Kan du fylla mitt glas med vatten från kranen? = Can you fill my glass with water from the tap?

If you want to sound a bit softer or more formal, you could say:

• Skulle du kunna fylla mitt glas med vatten från kranen? = Could you fill my glass with water from the tap?

But Kan du ... ? is very common and not rude by itself.

Yes, in practice it often covers both ability and request.

Literally, kan du means can you, but in everyday speech it is very often used to make a request:

• Kan du hjälpa mig? = Can you help me?
• Kan du öppna fönstret? = Can you open the window?

So in this sentence, it is not really asking about the person’s physical ability; it is functioning as a polite request.

Because the sentence is built in a fairly straightforward way:

• Kan du = can you
• fylla = fill
• mitt glas = my glass
• med vatten = with water
• från kranen = from the tap

So the structure is:

Kan du + verb + object + med-phrase + från-phrase

The phrase från kranen describes the water, not the glass. In other words, it means:

• fill my glass
• with water
• from the tap

Yes. Kranvatten means tap water.

So you could say:

• Kan du fylla mitt glas med kranvatten?

That is also natural. The difference is mainly one of style:

• med vatten från kranen = with water from the tap
• med kranvatten = with tap water

Both work, but the original sentence spells the idea out more directly.

A careful approximate pronunciation is:

Kahn doo FYL-lah mitt glahs med VAHT-ten frohn KRAH-nen?

A few notes:

• kan sounds roughly like kahn
• du is like doo
• fylla has a short y sound that English does not have exactly; it is not the same as filla or foola
• glas has a long a sound, roughly glahs
• vatten has a short a and double tt, so the vowel is short
• kranen is stressed on the first syllable: KRAH-nen

In normal speech, du may also sound less fully pronounced than doo.

Yes.

• fylla mitt glas = fill my glass
• fylla på mitt glas often means top up / refill my glass

So if the glass is empty, fylla mitt glas is fine. If the glass already has something in it and you want more added, fylla på mitt glas may be more natural.

Example:

• Kan du fylla på mitt glas? = Can you top up my glass?