Når jeg rydder ud i skabet, har jeg svært ved at skille mig af med gamle breve.

Når here means when in the sense of whenever / at the times that.

It introduces a time clause:

• Når jeg rydder ud i skabet = When I clear out the cupboard / Whenever I clean out the cupboard

This is different from some other time words in Danish:

• når = when, whenever, as soon as
• da = when, but usually about a specific event in the past
• mens = while

So in this sentence, når suggests a general situation, not just one single past occasion.

This is because Danish main clauses follow the V2 rule: the finite verb usually comes in the second position.

The sentence has:

1. a subordinate clause: Når jeg rydder ud i skabet
2. the main clause: har jeg svært ved at skille mig af med gamle breve

In the subordinate clause, the word order is normal:

• Når jeg rydder ud i skabet

But when that subordinate clause comes first, the main clause must invert:

• har jeg not
• jeg har

So:

• Når jeg rydder ud i skabet, har jeg svært ved ...

If you started with the main clause, you would get:

• Jeg har svært ved at skille mig af med gamle breve, når jeg rydder ud i skabet.

Both are grammatical, but the word order changes because of the V2 rule.

Rydde ud is a verb expression meaning to clear out, to declutter, or to get rid of things while cleaning up.

In infinitive form, it is:

• at rydde ud

In the present tense, the verb is conjugated and the particle stays separate:

• jeg rydder ud

This is similar to separable phrasal-style expressions in other languages. The main verb is rydder, and ud is a particle that helps give the meaning clear out rather than just tidy.

Compare:

• rydde = tidy, clear
• rydde ud = clear out, throw out unnecessary things

I skabet literally means in the cupboard / in the closet.

In this sentence, it goes with rydder ud in the sense of clearing out in the cupboard, meaning you are sorting through the contents of the cupboard.

Even though English often says clear out the cupboard, Danish can say:

• rydde ud i skabet

So the idea is not necessarily movement out of the cupboard in grammar terms, but the location where the clearing-out activity happens.

Skabet is the definite singular form: the cupboard / the closet.

Danish often adds the definite article to the end of the noun:

• et skab = a cupboard
• skabet = the cupboard

So:

• i skabet = in the cupboard

Using skab by itself here would be incomplete unless it had another determiner.

Har svært ved at means have difficulty doing something or find it hard to do something.

Structure:

• have svært ved at + infinitive

So:

• har svært ved at skille mig af med ... = have difficulty parting with ... = find it hard to get rid of ...

Literally, svært means difficult, and ved at is part of the fixed expression.

Some examples:

• Jeg har svært ved at sove. = I have trouble sleeping.
• Hun har svært ved at forstå det. = She has difficulty understanding it.

This is a fixed verbal expression:

• at skille sig af med noget

It means:

• to get rid of something
• to part with something
• to dispose of something

It has several parts:

• skille = separate
• sig = oneself
• af med = part of the fixed expression

So you should learn it as a whole:

• skille sig af med

In the sentence, sig changes to match the subject:

• jeg → mig
• du → dig
• han/hun/man → sig
• vi → os
• I → jer
• de → sig

That is why we get:

• at skille mig af med gamle breve

Because the subject is jeg.

The dictionary form is:

• at skille sig af med

But sig is really just the neutral reflexive form used for third person in dictionary entries. In actual sentences, the reflexive pronoun changes according to the subject:

• Jeg skiller mig af med det.
• Du skiller dig af med det.
• Hun skiller sig af med det.
• Vi skiller os af med det.

So in your sentence:

• jeg ... skille mig af med is correct because the subject is I.

Because ved at + infinitive is part of the construction after have svært.

The pattern is:

• have svært ved at + infinitive

So:

• har svært ved at skille mig af med ...

This is similar to English patterns like:

• have trouble doing
• find it hard to do

The at marks the infinitive skille.

Because brev is a neuter noun, and its plural is:

• et brev = a letter
• breve = letters

So:

• gamle breve = old letters

Also, the adjective gammel changes form here because it is describing a plural noun:

• gammel = old
• gamle = old, in plural and some definite contexts

Examples:

• et gammelt brev = an old letter
• gamle breve = old letters

So gamle breve is exactly the form you would expect.

Because Danish often uses the present tense for a general, repeated, or typical situation.

So this sentence means something like:

• whenever I clear out the cupboard, I find it hard to part with old letters

It is not necessarily about what is happening at this exact second. It can describe a habit or recurring experience.

English often does the same:

• When I clean out the cupboard, I have a hard time throwing away old letters.

Yes. Skab is a broad word.

Depending on context, et skab can mean:

• cupboard
• cabinet
• closet
• wardrobe

So i skabet could be:

• in the cupboard
• in the cabinet
• in the closet

The exact translation depends on what kind of storage furniture is meant.

Yes, it is natural and idiomatic.

Several parts are especially common and useful to learn as chunks:

• når ...
• rydde ud i ...
• have svært ved at ...
• skille sig af med ...

A learner will sound much more natural if they learn these as set patterns rather than translating word by word.