Snøen ligger dypt i hagen.

Snø means snow.
Snøen means the snow.

In Norwegian, the definite article (the) is usually attached to the end of the noun:

• snø = snow
• snøen = the snow
• hage = garden
• hagen = the garden

So Snøen ligger dypt i hagen literally is The snow lies deep in the garden.

Ligge = to lie (to be in a lying position).
Være = to be.

Norwegian often uses position verbs instead of være when talking about where something is:

• ligger – lies / is lying (for things that are flat, spread out, or on the ground)
• står – stands / is standing (for upright objects)
• sitter – sits / is sitting (for sitting position or “located” in some cases)

Snow is on the ground, spread out, so Norwegian naturally says:

• Snøen ligger dypt i hagen
  literally: The snow lies deep in the garden.

You can sometimes use er, but ligger sounds much more natural here.

The basic adjective is dyp = deep.

Dypt here is the adverb form, meaning deeply or deep (in depth). In Norwegian, adverbs are often formed by taking the neuter form of the adjective, which often ends in -t:

• dyp (common gender) → dypt (neuter / adverb)
• praktisk → praktisk (adverb – some don’t change)
• rask → raskt

In Snøen ligger dypt i hagen, dypt tells you how the snow is lying (the manner/degree), so it functions as an adverb.

Snøen er dyp i hagen is grammatically correct but sounds less natural and slightly different:

• Snøen ligger dypt i hagen – focuses on the state/extent of how the snow lies there. Very idiomatic.
• Snøen er dyp i hagen – sounds more like “The snow is deep in the garden” as an inherent property; it can work, but Norwegians prefer ligger dypt for snow on the ground.

So: use ligger dypt in normal speech.

Yes. Both are correct:

• Snøen ligger dypt i hagen.
• I hagen ligger snøen dypt.

Norwegian main clauses follow the V2 rule: the verb must be in second position. In the second version:

1. I hagen (adverbial/fronted phrase)
2. ligger (verb – second position)
3. snøen dypt (rest of the sentence)

I hagen ligger snøen dypt puts extra emphasis on in the garden (contrasting with other places).

I usually means in, inside, within a space.
På usually means on, on top of, on a surface (though there are many idiomatic uses).

For areas like gardens, parks, forests, etc., Norwegian typically uses i:

• i hagen – in the garden
• i parken – in the park
• i skogen – in the forest

So Snøen ligger dypt i hagen sounds natural; på hagen would sound wrong in this context.

Same reason as with snø / snøen:

• hage = garden
• hagen = the garden

I hagen therefore means in the garden (a specific one that both speaker and listener know about).

If you said i en hage, it would mean in a garden (some garden, not specified).

Singular:

• hage = a garden
• hagen = the garden

Plural:

• hager = gardens
• hagene = the gardens

So:

• Snøen ligger dypt i hagene.
  = The snow lies deep in the gardens. (specific gardens)

• Snøen ligger dypt i hager.
  = The snow lies deep in gardens. (some unspecified gardens; sounds a bit abstract/general)

An approximate IPA transcription (Eastern Norwegian):

• Snøen: [ˈsnøː.ən]
  
  • ø: like the vowel in British “sir” or French “peu”, but rounded
• ligger: [ˈlɪɡər]
• dypt: [dyːpt]
  
  • y: like German ü in “Tür”; say ee in “see” while rounding your lips
• i: [i] (like English ee in “see”)
• hagen: [ˈhɑː.ɡən]
  
  • a: like a in British “father”

Put together roughly:
[ˈsnøː.ən ˈlɪɡər dyːpt i ˈhɑː.ɡən]

Ligger is the present tense of å ligge (to lie).

• Snøen ligger dypt i hagen.
  = The snow lies / is lying deep in the garden. (now)

Past tense is lå:

• I går lå snøen dypt i hagen.
  = Yesterday the snow lay / was lying deep in the garden.

Norwegian usually does not have a separate continuous/progressive tense like English “is lying”, “was lying”.

The simple present covers both:

• Snøen ligger dypt i hagen.
  = The snow lies deep in the garden.
  = The snow is lying deep in the garden.

Context tells you which English translation fits better. You don’t need to add anything extra in Norwegian.

You can modify dypt with an adverb:

• Snøen ligger veldig dypt i hagen. – The snow is very deep in the garden.
• Snøen ligger utrolig dypt i hagen. – The snow is incredibly deep in the garden.
• Snøen ligger kjempedypt i hagen. – The snow is really deep in the garden (informal).

Comparative and superlative of dyp:

• dyp – deep
• dypere – deeper
• dypest – deepest

Example:

• Her ligger snøen dypere enn i gata. – Here the snow lies deeper than in the street.