Jeg beholder mobilnummeret kun hvis nummerflytningen er gratis.

Beholder is the present tense of beholde, which means to keep, to retain, or to hold on to.

So:

• jeg beholder = I keep / I am keeping

In Danish, the present tense is very often made by adding -r to the infinitive:

• at beholde = to keep
• beholder = keep / keeps / am keeping

Unlike English, Danish does not change the verb for different persons:

• jeg beholder = I keep
• du beholder = you keep
• han/hun beholder = he/she keeps

Because mobilnummeret is the definite form, meaning the mobile number.

• et mobilnummer = a mobile number
• mobilnummeret = the mobile number

In this sentence, Danish uses the definite form because it refers to a specific number already understood from context.

Also, mobilnummer is a compound noun:

• mobil = mobile
• nummer = number

The gender of the whole compound comes from the last part, nummer, which is a neuter noun, so the definite ending is -et:

• et nummer
• nummeret
• therefore et mobilnummer
• mobilnummeret

Because Danish, like German, very often makes compound nouns by writing them as one word.

So:

• nummer = number
• flytning = moving / transfer
• nummerflytning = number transfer / number porting

Then the definite form becomes:

• nummerflytningen = the number transfer / the number porting

Writing it as separate words would usually be wrong in Danish. Compound nouns are normally written together.

Here nummerflytningen means the transfer of the phone number from one provider to another. In telecom English, this is often called number porting or number transfer.

So the sentence is talking about keeping the same mobile number, but only if moving that number to the new provider costs nothing.

Grammatically:

• en flytning = a move / a transfer
• flytningen = the move / the transfer

Because flytning is a common-gender noun, the definite ending is -en, which is why you get nummerflytningen.

Kun hvis means only if, not just if.

• hvis = if
• kun hvis = only if

That makes the condition stronger.

Compare:

• Jeg beholder mobilnummeret hvis nummerflytningen er gratis.
  = I keep the mobile number if the transfer is free.

• Jeg beholder mobilnummeret kun hvis nummerflytningen er gratis.
  = I keep the mobile number only if the transfer is free.

The second version means that being free is the required condition.

Yes, the position of kun can change what it emphasizes.

In your sentence:

• Jeg beholder mobilnummeret kun hvis nummerflytningen er gratis.

Here kun naturally goes with hvis, so it means only if.

If you moved kun, the focus could change. For example:

• Jeg beholder kun mobilnummeret ...
  = I only keep the mobile number ...
  This could suggest that the mobile number is the only thing I keep.

So the original placement is good because it clearly expresses the condition:

• only if the number transfer is free

Because hvis introduces a subordinate clause, and subordinate clauses in Danish do not use normal main-clause inversion.

In a main clause, Danish typically follows the verb-second pattern:

• Nummerflytningen er gratis.
  = The number transfer is free.

But after hvis, the clause becomes subordinate, and the subject normally comes before the verb:

• hvis nummerflytningen er gratis
  = if the number transfer is free

So this is the expected Danish word order.

Yes, you could, depending on context.

• mobilnummeret = the mobile number
• mit mobilnummer = my mobile number

In many contexts, mobilnummeret is enough because it is already obvious whose number is meant. Danish often uses the definite form where English might prefer a possessive.

So both can be possible:

• Jeg beholder mobilnummeret ...
• Jeg beholder mit mobilnummer ...

The version with mit is more explicit. The version with -et sounds more neutral and context-based.

Because gratis is normally used as an indeclinable adjective/adverb in modern Danish. It stays the same form.

So you say:

• Det er gratis. = It is free.
• Nummerflytningen er gratis. = The number transfer is free.

You do not change it for gender or number the way you do with many other adjectives.

For example, with a regular adjective:

• Det er dyrt. = It is expensive.
• Varen er dyr. = The item is expensive.

But with gratis, the form stays gratis.

Yes, it is natural and understandable Danish.

A Dane might also say it in slightly different ways, for example:

• Jeg beholder mobilnummeret, hvis nummerflytningen er gratis.
• Jeg beholder kun mobilnummeret, hvis det er gratis at flytte det.
• Jeg beholder mit mobilnummer, men kun hvis nummerflytningen er gratis.

These are just stylistic variations. Your original sentence is perfectly good Danish and clearly communicates the idea.