Døren kan ikke åbnes, fordi håndtaget sidder løst, og låsen virker heller ikke altid.

Because Danish often marks the as a suffix on the noun.

• dør = door → døren = the door
• håndtag = handle → håndtaget = the handle
• lås = lock → låsen = the lock

You can also see noun gender here:

• dør and lås are common gender, so they usually take -en in the definite singular.
• håndtag is neuter, so it takes -et.

So this sentence is using definite nouns because it is talking about a specific door, handle, and lock.

Åbnes is a passive form.

• åbne = to open
• åbnes = be opened / open in the passive sense

So Døren kan ikke åbnes means literally something like The door cannot be opened.

This is a very common Danish pattern:

• kan åbne = can open
• kan åbnes = can be opened

Danish often uses the -s passive where English uses be + past participle.

Because in a Danish main clause, the finite verb normally comes early, and ikke usually comes after that finite verb.

Here the finite verb is kan, so the normal order is:

• Døren kan ikke åbnes

That is the standard main-clause pattern.

A useful contrast:

• Main clause: Døren kan ikke åbnes
• Subordinate clause: ..., fordi døren ikke kan åbnes

So the position of ikke depends on the type of clause.

Fordi means because and introduces a subordinate clause.

In subordinate clauses, Danish does not use the usual main-clause V2 pattern. The difference becomes especially clear when adverbs like ikke are present.

Compare:

• Main clause: Låsen virker ikke altid
• Subordinate clause: ..., fordi låsen ikke altid virker

In your sentence, håndtaget sidder løst does not visibly look very different from English word order, but it is still part of a subordinate structure introduced by fordi.

This is an idiomatic and very common use of sidde in Danish.

Danish often uses sidde for something that is attached, positioned, or fitted somewhere.

So:

• sidder fast = is firmly attached / stuck
• sidder løst = is loose

With objects like handles, screws, buttons, plugs, and similar things, sidde is very natural. So håndtaget sidder løst is the normal way to say that the handle is loose.

Heller ikke means something like not either, also not, or doesn't ... either.

So:

• låsen virker ikke altid = the lock does not always work
• låsen virker heller ikke altid = the lock does not always work either

The phrase links this problem to the previous one. First the handle is loose, and in addition the lock also has a problem.

English often expresses this with either:

• The lock doesn’t always work either.

Because that is a natural Danish adverb order.

In a main clause, the finite verb comes first among the verbal elements, and adverbs then follow in a fairly regular order. Here:

• virker = finite verb
• heller ikke = sentence adverbial
• altid = frequency adverb

So låsen virker heller ikke altid is a very natural order.

Also note the nuance:

• ikke altid means not always, not never

So the lock works sometimes, but not consistently.

Yes, often you could.

• virker = works
• fungerer = functions / works

Both can be correct, but virker is often the more everyday, natural choice for whether a device, mechanism, or lock works.

So:

• låsen virker ikke altid sounds very natural
• låsen fungerer ikke altid is also possible, but can sound a little more formal or technical

Because og is joining two separate clauses, and the second clause has its own subject.

The two clauses are:

• håndtaget sidder løst
• låsen virker heller ikke altid

Since the subject changes from håndtaget to låsen, Danish must state the new subject. You cannot leave it out here.

So the structure is:

• ..., fordi håndtaget sidder løst, og låsen virker heller ikke altid.

That is completely normal Danish.