Ikke dytt døren.

Because this is a command. In Norwegian, as in English, commands usually leave out the subject.

• Dytt døren. = Push the door.
• Ikke dytt døren. = Don’t push the door.

The understood subject is you.

Dytt is the imperative form of the verb å dytte.

A very common pattern in Norwegian is:

• infinitive: å dytte
• imperative: dytt!

So here, dytt means push!

Ikke means not.

When you put it with an imperative, it makes a negative command:

• Dytt døren. = push the door
• Ikke dytt døren. = don’t push the door

In short warnings and instructions, putting ikke first is very common:

• Ikke rør! = don’t touch!
• Ikke løp! = don’t run!

Because døren is the definite singular form: the door.

Norwegian usually adds the definite article to the end of the noun:

• en dør = a door
• døren = the door

So the sentence refers to a specific door, not just any door.

Yes. In Bokmål, you may see both:

• døren
• døra

Both can mean the door.

Why? Because dør is a noun that can often be treated with either a common-gender ending or a feminine ending in Bokmål:

• en dør → døren
• ei dør → døra

Døren is often a bit more neutral in writing, while døra is very common in everyday speech.

Yes. Et dytt means a push or a nudge.

So dytt can be:

• a verb form: Dytt! = Push!
• a noun: et dytt = a push

In Ikke dytt døren, it is clearly a verb, because it is part of a command.

Yes, there can be a small difference.

• dytte døren = push the door
• dytte på døren = push on the door / push against the door

Without på, the door is the direct object.
With på, the wording focuses a little more on the physical action against the door.

In many situations, both are possible, but they are not always equally natural in every context.

No. Norwegian has several verbs that can overlap in meaning, for example:

• å dytte = push, shove, nudge
• å skyve = push, move by pushing
• å presse = press, push with pressure

So depending on context, a Norwegian speaker might also use another verb.
Dytt is completely grammatical here, but another verb might sound more natural in a specific real-life situation.

A rough English-friendly guide is:

• ikke ≈ IK-keh
• dytt = the y is not like English y; it is a rounded vowel, similar to German ü or French u
• døren = DUR-en with a rounded vowel; ø is somewhat like German ö or French eu

The hardest sounds for English speakers are usually y and ø, because English does not really have exact matches for them.