Hun rydder altanen, mens jeg tømmer de sidste kasser i gangen.

They are related in meaning, but they are not interchangeable.

• rydde means to tidy up, clear away, put in order
• tømme means to empty out

So:

• Hun rydder altanen = she is tidying/clearing the balcony
• jeg tømmer de sidste kasser = I am emptying the last boxes

If you used tømmer altanen, it would sound more like you are removing everything from the balcony. If you used rydder kasserne, it would sound more like you are sorting/putting away the boxes, not specifically emptying them.

Because they are both in the present tense.

The infinitives are:

• at rydde = to tidy / to clear
• at tømme = to empty

Present tense in Danish is usually formed by adding -r to the infinitive:

• rydde → rydder
• tømme → tømmer

In this sentence, Danish uses the present tense for actions happening right now, where English often uses the present progressive:

• Hun rydder altanen = She is tidying the balcony
• jeg tømmer... = I am emptying...

So Danish present tense often covers both English simple present and is/are + -ing.

Because altanen means the balcony, while altan means a balcony or just balcony as a dictionary form.

Danish usually puts the definite article on the end of the noun:

• en altan = a balcony
• altanen = the balcony

This is very common in Danish:

• en gang = a hallway
• gangen = the hallway

So the English word the is often built into the noun itself in Danish.

Because Danish normally uses the attached definite ending on the noun by itself:

• altanen = the balcony

You use den/det/de together with a noun when there is usually an adjective before it:

• den store altan = the big balcony
• de sidste kasser = the last boxes

So:

• altanen = correct
• den altan = only in special contexts, such as contrast: ikke den altan, men den anden = not that balcony, but the other one

Here mens means while.

It connects two actions that happen at the same time:

• Hun rydder altanen, mens jeg tømmer de sidste kasser i gangen.
• She tidies the balcony while I empty the last boxes in the hallway.

Mens can sometimes also have a contrastive sense like whereas, but in this sentence it is clearly about simultaneous actions.

Because Danish definite noun phrases with an adjective usually use this pattern:

de/den/det + adjective + noun

So:

• de sidste kasser = the last boxes

This is the normal way to say it.

Compare:

• kasser = boxes
• kasserne = the boxes
• de sidste kasser = the last boxes

You do not normally combine the adjective with a definite noun ending in the way English speakers might expect, so sidste kasserne is not standard Danish here.

I gangen means in the hallway, and that is a very natural choice because a hallway is treated as an interior space.

• i gangen = in the hallway
• i køkkenet = in the kitchen
• i stuen = in the living room

You may also hear på gangen in some contexts, especially in places like schools, hospitals, or offices, where it can mean something like out in the corridor. But in this sentence, i gangen is a very normal and neutral choice.

Because mens introduces a subordinate clause, and Danish subordinate clauses normally keep the order subject + verb:

• mens jeg tømmer...

In a main clause, Danish often puts the verb in the second position, but after a subordinating word like mens, the structure is more straightforward.

So this is normal:

• Hun rydder altanen, mens jeg tømmer de sidste kasser i gangen.

If you reverse the order and put the mens-clause first, then the main clause shows inversion:

• Mens jeg tømmer de sidste kasser i gangen, rydder hun altanen.

Notice that after the opening clause, the main clause becomes rydder hun, not hun rydder.

Yes, it is very common and correct to put a comma there, because mens jeg tømmer de sidste kasser i gangen is a subordinate clause.

So this version is normal:

• Hun rydder altanen, mens jeg tømmer de sidste kasser i gangen.

You may sometimes see variation in comma use in Danish depending on comma style, but for a learner, this comma is a good and natural one to use.

Because kasser is the plural form of en kasse.

The pattern is:

• en kasse = a box
• kasser = boxes
• kasserne = the boxes

In this sentence, de sidste kasser means the last boxes, so the plural makes sense because there is more than one box left.

The ø sound does not have an exact English equivalent, which is why English speakers often find it tricky.

A rough guide:

• ø is a rounded front vowel
• it is somewhat similar to the vowel in French deux or German schön
• you make a sound a bit like English eh/uh, but with your lips rounded

So tømmer is roughly pronounced something like TUM-er or TEU-mer, but neither of those is exact. The important thing is that the vowel is fronted and rounded.

If you want to sound more natural, it helps to listen to native speakers and repeat the word several times:

• tømme
• tømmer

Yes. Rydde is flexible.

Depending on context, rydde altanen can suggest:

• tidying the balcony
• clearing the balcony
• putting things away on the balcony
• making the balcony neat

So the exact English translation depends on the situation. If she is sweeping, organizing, and removing clutter, tidying the balcony is a good translation. If she is removing things from it so it becomes empty or usable, clearing the balcony may be better.