Hvis batterierne i fjernbetjeningen er gamle, må jeg skifte dem.

Hvis is used for a condition: if X happens, then Y happens.

So Hvis batterierne i fjernbetjeningen er gamle... means If the batteries in the remote control are old...

By contrast, om is usually used for indirect yes/no questions, as in I don’t know if they are old.

The ending -ne makes the noun definite plural, meaning the batteries.

Here is the pattern:

• et batteri = a battery
• batteriet = the battery
• batterier = batteries
• batterierne = the batteries

So batterierne means we are talking about specific batteries, namely the ones in the remote control.

Danish usually writes compound nouns as one word.

So:

• fjern = remote / distant
• betjening = operation / control
• fjernbetjening = remote control

Then -en is added to make it definite:

• en fjernbetjening = a remote control
• fjernbetjeningen = the remote control

This is very normal in Danish.

Because the batteries are physically inside the remote control.

• i = in / inside
• på = on

So:

• batterierne i fjernbetjeningen = the batteries in the remote control
• knapperne på fjernbetjeningen = the buttons on the remote control

The adjective changes form depending on the noun.

With gammel:

• en gammel fjernbetjening = an old remote control
• et gammelt batteri = an old battery
• gamle batterier = old batteries

Since batterierne is plural, the adjective must be gamle.

So:

• batterierne er gamle = the batteries are old

Because Danish main clauses usually follow the verb-second rule.

When a subordinate clause comes first, it takes the first position, and then the finite verb comes next.

So:

• Jeg må skifte dem. = I may/must replace them.
• Hvis batterierne i fjernbetjeningen er gamle, må jeg skifte dem.

The first whole chunk is Hvis batterierne i fjernbetjeningen er gamle, so the next thing must be the verb må, not the subject jeg.

Må is a modal verb. It can mean either:

• may / am allowed to
• must / have to

The exact meaning depends on context.

So by itself, må jeg skifte dem could mean either:

• May I replace them?
• I must replace them

In a full sentence, the surrounding context usually makes the intended meaning clear.

After modal verbs such as må, kan, skal, vil, Danish uses the bare infinitive without at.

So you say:

• jeg må skifte dem
• jeg kan gøre det
• jeg vil prøve

Not:

• jeg må at skifte dem

This is similar to English I must replace them, not I must to replace them.

Because dem is the object form of they.

• de = they
• dem = them

Compare:

• De er gamle. = They are old.
• Jeg skifter dem. = I replace them.

In your sentence, the batteries are receiving the action of skifte, so Danish uses dem.

Both are possible.

• skifte often means change or replace
• udskifte more clearly emphasizes replace with another/new one

So:

• må jeg skifte dem = I may/must change them
• må jeg udskifte dem = I may/must replace them

The original sentence is perfectly natural, but udskifte can sound a bit more explicit.

Yes, it is grammatical and natural.

However, if you specifically mean batteries that no longer work well, Danish often uses more precise words such as:

• flade batterier = dead/flat batteries
• brugte batterier = used batteries

So gamle is fine if you really mean old, but if the idea is run-down or dead, Danish may prefer a more specific word.

Because the sentence starts with a subordinate clause:

• Hvis batterierne i fjernbetjeningen er gamle

Then the main clause follows:

• må jeg skifte dem

The comma marks the boundary between those two parts. In standard written Danish, that comma is expected here.