De linker deur van de kast is open.

No. In this sentence, linker does not mean a comparative like more left.

Here, linker is the normal form used before a noun to mean left-hand or on the left:

• de linker deur = the left door
• mijn linkerhand = my left hand

The form links is usually used in other positions, for example:

• de deur links = the door on the left
• links van de kast = to the left of the cupboard

So de linker deur is the natural way to say the left door before the noun.

Because linker is an adjective describing the noun deur.

In Dutch, adjectives are normally written separately from the noun:

• de linker deur
• een grote kast
• de open deur

Dutch often does make long compound words, but those are usually noun + noun combinations, such as:

• kastdeur = cupboard door

So linker deur stays as two words because it is adjective + noun.

Because both deur and kast are de-words in Dutch.

Dutch singular nouns normally take either de or het:

• de deur
• de kast
• het huis
• het raam

There is no special reason in this sentence beyond the fact that both nouns happen to belong to the de group.

Van de kast means of the cupboard / cabinet.

It tells you which door is being talked about: not just any left door, but the left door of the cupboard.

So the structure is:

• de linker deur = the left door
• van de kast = of the cupboard

Together:

• de linker deur van de kast = the left door of the cupboard

Yes.

Dutch often uses a compound noun where English might use of:

• de deur van de kast
• de kastdeur

Both are possible, though the nuance can be slightly different.

• de deur van de kast sounds a bit more explicit: the door of the cupboard
• de kastdeur is a compact noun: the cupboard door

So you could also say:

• De linker kastdeur is open.

That is perfectly normal Dutch.

Because van de kast belongs with the noun phrase de linker deur.

It is part of the subject, not part of the predicate.

So the sentence is built like this:

• De linker deur van de kast = the full subject
• is = the verb
• open = the description

Dutch keeps the noun and its modifier together, so van de kast stays next to deur.

Because open describes the state of the door: it is not closed.

• De deur is open. = The door is open.

Geopend is the past participle of openen and often sounds more like has been opened or more formal/passive:

• De deur is geopend. = The door has been opened / The door is opened.

For an ordinary description of a door’s condition, open is much more natural.

Because it comes after the verb is.

In Dutch, adjectives usually take an ending when they come before a noun, but not when they are used after a linking verb like zijn:

• de open deur = the open door
• de deur is open = the door is open

So:

• before the noun: often an ending
• after is: usually no ending

That is why the sentence has open, not opene.

Yes, but the meaning or emphasis changes a little.

• De deur links van de kast is open means the door to the left of the cupboard is open.
  That is a different structure: now the door is next to the cupboard, not a door of the cupboard.

• De deur links is open can mean the door on the left is open.
  This is similar in meaning to de linker deur, but phrased differently.

• De linker deur staat open is also very natural Dutch.
  With doors, windows, and gates, Dutch often uses staan open to describe their open position.

So De linker deur van de kast is open is correct, but there are other natural ways to express related ideas.