Återanvänder du hellre matlådan än att köpa en ny?

Because this is a yes/no question. In Swedish, the finite verb usually comes first in yes/no questions, followed by the subject:

• Återanvänder du ...? = Do you reuse ...?

Swedish does not use a separate helper verb like English do here. Instead of Do you reuse...?, Swedish simply moves the main verb to the front.

Återanvänder is the present tense of återanvända, which means to reuse.

It is built from:

• åter- = again / back / re-
• använda = to use

So återanvända literally means use again, but in normal English reuse is the best translation.

Återanvänder can mean reuse, are reusing, or do reuse, depending on context.

That is just how Swedish works. English uses do to form many questions, but Swedish does not.

So:

• Do you reuse ...?
  becomes
• Återanvänder du ...?

The Swedish verb återanvänder is in the present tense, and that alone is enough.

Hellre means rather or preferably. It shows that one option is preferred over another.

Examples:

• Jag går hellre. = I’d rather walk.
• Hon stannar hellre hemma. = She would rather stay home.

In your sentence, hellre shows a preference for reusing the lunch box instead of buying a new one.

Än means than when you compare two things or two actions.

After hellre, Swedish often uses this pattern:

• hellre ... än ... = rather ... than ...

Here, the second part is an action, so Swedish uses an infinitive phrase:

• att köpa = to buy

So än att köpa en ny means than to buy a new one.

In natural English, we often leave out to after rather and say rather reuse it than buy a new one, but Swedish commonly keeps att in this kind of structure.

No. A very common alternative is:

• i stället för att köpa en ny = instead of buying a new one

So a similar sentence could be:

• Återanvänder du hellre matlådan i stället för att köpa en ny?

That may sound a little more straightforward to some learners. Both patterns are useful to know.

Matlådan is the definite singular form:

• en matlåda = a lunch box / a food container
• matlådan = the lunch box / the food container

The definite form suggests that the speaker is talking about a specific lunch box that is already known from the situation or conversation.

Matlåda literally means food box, but in everyday Swedish it often means:

• a lunch box
• a food container
• a packed meal you bring with you

It can refer to the container itself, or sometimes the meal in it, depending on context. In many situations, lunch box or meal-prep container is a good English match.

Because matlåda is an en-word.

In Swedish, adjectives must agree with the noun:

• en ny matlåda
• ett nytt glas

Since matlåda takes en, the adjective is ny, not nytt.

Because the noun is already understood from the context.

• köpa en ny = buy a new one

This is very common in Swedish. If the listener already knows what noun is meant, the noun can be left out.

Other examples:

• Jag tar den röda. = I’ll take the red one.
• Hon vill ha en större. = She wants a bigger one.

So en ny here really means en ny matlåda.

Sometimes, but återanvänder is better here.

• återanvända = reuse
• använda igen = use again

Återanvända has a clearer sense of reusing something instead of replacing it, which fits this sentence very well. Använda igen is more general and does not always have that same reuse feeling.

Yes, it is understandable and grammatical.

A few natural alternatives are:

• Återanvänder du hellre matlådan än att köpa en ny?
• Återanvänder du hellre matlådan i stället för att köpa en ny?
• Föredrar du att återanvända matlådan i stället för att köpa en ny?

The original sentence is fine. The alternatives just show other common ways to express the same idea.