Min taske er åben, fordi lynlåsen er gået i stykker.

Because min already means my, and in Danish a possessive already makes the noun definite.

So you say:

• min taske = my bag
• tasken = the bag

You do not combine a possessive with the definite ending -en here, so min tasken is incorrect.

Because taske is a common gender noun: en taske.

When an adjective comes after er, it still agrees with the noun:

• en taske er åben
• et vindue er åbent
• to tasker er åbne

So in this sentence, åben is the correct form because taske is common gender singular.

Lynlåsen means the zipper.

The basic noun is:

• en lynlås = a zipper

The definite singular form is:

• lynlåsen = the zipper

It is also a compound word:

• lyn = lightning
• lås = lock

So Danish literally builds a word something like lightning-lock.

Danish often uses the definite form when the thing is already understood from the context.

Here, once we hear my bag, the zipper being talked about is obviously the zipper of the bag. So lynlåsen sounds natural.

You could say min lynlås in some situations, but it would usually be used only if you really want to stress ownership or contrast it with someone else’s zipper.

Gå i stykker is a very common Danish expression meaning to break or to stop working.

Literally, it is something like go into pieces, but in normal use it is just the standard way to say that an object has become broken.

Examples:

• Min telefon er gået i stykker. = My phone has broken.
• Brillerne gik i stykker. = The glasses broke.
• Vaskemaskinen er gået i stykker. = The washing machine has broken.

It is used for many kinds of objects, not only things that literally fall into pieces.

In this expression, Danish normally uses er in the perfect tense:

• er gået i stykker

This is because gå in expressions about movement or change of state often takes være in the perfect.

So:

• Lynlåsen er gået i stykker. = The zipper has broken.

This is something you mainly learn as a pattern. It is one of those common combinations that is best remembered as a whole phrase.

Because er gået i stykker focuses on the result that is relevant now.

• Lynlåsen gik i stykker. = The zipper broke.
  This talks about a past event.

• Lynlåsen er gået i stykker. = The zipper has broken / is broken.
  This connects the past event to the present situation.

Since the bag is open now because the zipper is broken now, the perfect tense is very natural.

No, not quite.

• åben = open, describing a state
• åbnet = opened, a past participle

So:

• Min taske er åben. = My bag is open.
• Min taske er blevet åbnet. = My bag has been opened.

In your sentence, åben is right because it describes the bag’s current state, not the action of opening it.

Yes. Fordi introduces a subordinate clause, and Danish subordinate clauses follow different word order from main clauses.

In your sentence, you do not really notice it, because there is no word like ikke to make the difference visible:

• ... fordi lynlåsen er gået i stykker

But compare:

• main clause: Lynlåsen virker ikke.
• subordinate clause: ... fordi lynlåsen ikke virker.

So after fordi, sentence adverbs such as ikke usually come before the finite verb.

Sometimes yes, because for can also mean because in Danish.

So you may hear:

• Min taske er åben, for lynlåsen er gået i stykker.

But there is a difference:

• fordi introduces a subordinate clause
• for introduces a coordinating clause

For a learner, fordi is usually the safest and most neutral choice when you want to say because.