Vinterdekket er ødelagt, så vi må kjøpe et nytt.

Because Norwegian usually forms compound nouns as a single word.
So vinter + dekk becomes vinterdekk.

This is very common in Norwegian:

• sommerdekk = summer tire
• vinterdekk = winter tire
• bilnøkkel = car key

The last part, dekk, is the main noun, so it controls the gender and the endings.

In Norwegian, the definite article is usually attached to the end of the noun instead of being a separate word.

So:

• et dekk = a tire
• dekket = the tire

And with the compound:

• et vinterdekk = a winter tire
• vinterdekket = the winter tire

The ending -et is the definite ending for a neuter singular noun.

Because dekk is a neuter noun, so it takes:

• the article et
• the neuter form of the adjective, nytt

Compare:

• en ny bil = a new car
• et nytt dekk = a new tire

Since et nytt refers back to dekk, it has to stay neuter.

Adjectives in Norwegian often change form to match the gender and number of the noun.

For ny, the main forms are:

• ny for common gender singular
• nytt for neuter singular
• nye for plural and usually definite forms

So:

• en ny stol
• et nytt dekk
• nye dekk

That is why the sentence has et nytt.

Because the noun is understood from the context.
Et nytt here means a new one, where one refers to tire.

Norwegian does this just like English:

• Vi må kjøpe et nytt. = We have to buy a new one.
• Vi må kjøpe et nytt dekk. = We have to buy a new tire.

Both are possible, but leaving out dekk sounds natural when it is already clear.

Because after a modal verb like må, Norwegian uses the bare infinitive.

So you say:

• vi må kjøpe

not:

• vi må å kjøpe

This works much like English:

• we must buy not
• we must to buy

The same happens with other modal verbs such as kan, skal, vil, and bør.

Here så means so, introducing a result or consequence.

So the structure is:

• Vinterdekket er ødelagt = first statement
• så vi må kjøpe et nytt = result

In other words: the tire is damaged, so we have to buy a new one.

Because this så is a coordinating conjunction linking two main clauses. After a coordinating conjunction, Norwegian normally keeps regular main-clause word order:

• så vi må kjøpe et nytt

That means subject + verb:

• vi må

So there is no inversion here.

Here ødelagt is used like an adjective meaning broken or damaged.

It comes from a past participle, but in this sentence it describes the current state of the tire:

• dekket er ødelagt = the tire is damaged

You will often see it after er:

• Bilen er ødelagt
• Døra er ødelagt
• Vinterdekket er ødelagt

Usually it can correspond to both. In this sentence, English would most naturally say either:

• we must buy a new one
• we have to buy a new one

So må often covers both ideas of necessity.