Jeg kæmper med kufferten.

Because jeg is the subject form of I.

• Jeg = I
• Mig = me

In Jeg kæmper med kufferten, the speaker is the one doing the action, so Danish uses jeg.

Compare:

• Jeg kæmper = I struggle
• Hun hjælper mig = She helps me

Kæmpe is the infinitive form, meaning to struggle or to fight.

In a normal present-tense sentence, Danish usually adds -r to the infinitive:

• at kæmpe = to struggle
• jeg kæmper = I struggle / I am struggling

So kæmper is the present tense form.

It can mean both, depending on context.

Danish present tense often covers both:

• a general meaning: I struggle
• a right-now meaning: I am struggling

So Jeg kæmper med kufferten can describe:

• a general difficulty with the suitcase, or
• something happening at this moment

Context tells you which one is meant.

Med usually means with, but here it is part of the natural Danish expression kæmpe med, which means struggle with.

So:

• kæmpe med noget = to struggle with something

You normally need med here. Danish does not usually say this with a direct object alone.

• Jeg kæmper med kufferten = correct
• Jeg kæmper kufferten = not natural

Because kufferten is the definite form: the suitcase.

In Danish, the definite article is often added to the end of the noun:

• en kuffert = a suitcase
• kufferten = the suitcase

So the ending -en here means the.

That depends on the noun’s gender.

Danish has two main genders:

• common gender: usually takes en in the indefinite form and -en in the definite form
• neuter gender: usually takes et in the indefinite form and -et in the definite form

For this noun:

• en kuffert = a suitcase
• kufferten = the suitcase

So kuffert is a common gender noun.

Yes, very much so.

It is a common and natural way to say that someone has difficulty with something:

• Jeg kæmper med lektierne = I’m struggling with the homework
• Han kæmper med sproget = He’s struggling with the language
• Vi kæmper med problemet = We’re struggling with the problem

So in your sentence, kæmper med should be learned as a useful pattern.

Yes.

It can be:

• literal: you are physically struggling with a suitcase
• figurative: less likely here, but still possible depending on context

With kufferten, the most natural reading is physical: maybe the suitcase is heavy, awkward, or hard to move.

A rough pronunciation guide is:

• Jeg sounds roughly like yai or yaiy, depending on accent
• kæmper sounds roughly like KEM-per
• kufferten sounds roughly like KOO-fair-den or KOO-fuh-den, depending on how reduced the middle syllable is

About æ:

• æ is a vowel somewhat like the e in bed, but often a bit more open

A few pronunciation notes:

• Danish pronunciation varies quite a bit by region
• unstressed syllables are often reduced
• the r can affect the sound of nearby vowels

If you want to sound natural, listening and repeating is especially important with Danish.

Because Danish often puts the at the end of the noun instead of using a separate word in front.

So:

• English: the suitcase
• Danish: kufferten

That -en ending already includes the idea of the.

A separate word like den is used only in certain situations, such as when there is an adjective:

• kufferten = the suitcase
• den tunge kuffert = the heavy suitcase

You usually put the verb before the subject:

• Jeg kæmper med kufferten. = I am struggling with the suitcase.
• Kæmper jeg med kufferten? = Am I struggling with the suitcase?

This is a very common Danish pattern:

• statement: subject + verb
• yes/no question: verb + subject

The past tense of kæmpe is kæmpede.

So:

• Jeg kæmper med kufferten. = I struggle / I’m struggling with the suitcase.
• Jeg kæmpede med kufferten. = I struggled / I was struggling with the suitcase.

This is useful because many learners first meet the present tense and then want to know how to talk about the same situation in the past.

Yes, it is completely natural.

It sounds like something a Danish speaker would say if they were having trouble handling a suitcase.

It is short, idiomatic, and grammatically straightforward:

• Jeg = subject
• kæmper = present-tense verb
• med kufferten = prepositional phrase showing what the struggle is with

So it is a very good model sentence for learning both present tense and the pattern kæmpe med noget.