Kan du give beskeden videre til min søster?

Because this is a question.

In Danish, yes/no questions usually use verb-first word order:

• Du kan give beskeden videre. = You can pass the message on.
• Kan du give beskeden videre? = Can you pass the message on?

So kan moves in front of du to make it a question.

Because kan is a modal verb, and after a modal verb in Danish, the next verb normally stays in the infinitive.

So:

• kan give = can give
• not kan giver

This is similar to English:

• can give
• not can gives

Other common Danish modals work the same way:

• vil give = wants to give / will give
• skal give = must / is going to give
• må give = may / is allowed to give

Besked means message or information, depending on context.

The ending -en makes it definite:

• en besked = a message
• beskeden = the message

Danish often adds the definite article to the end of the noun instead of using a separate word like the.

So give beskeden videre literally contains the message, not a message.

Here videre means something like on, along, or further.

In the expression give noget videre, it means:

• pass something on
• relay something

So give beskeden videre is a very natural way to say pass the message on.

By itself, videre often has the idea of continuing or moving onward:

• gå videre = go on / continue
• læse videre = continue reading / continue studying

Because til marks the recipient here: the person the message is being passed to.

So:

• til min søster = to my sister

Danish often uses til where English uses to:

• sige noget til nogen = say something to someone
• give noget til nogen = give something to someone

In this sentence, til min søster tells you who should receive the message.

Yes, that is also possible.

Both of these are natural:

• Kan du give beskeden videre til min søster?
• Kan du give min søster beskeden videre?

The difference is mostly about word order and emphasis, not basic meaning.

• give beskeden videre til min søster keeps the message together with videre
• give min søster beskeden videre puts the recipient earlier

Both are understandable and normal.

Because søster is a common gender noun in Danish.

Danish has two grammatical genders:

• common gender: takes en, min, din, etc.
• neuter gender: takes et, mit, dit, etc.

So:

• en søster
• min søster

Compare:

• en bil → min bil
• et hus → mit hus

Since søster is an en-word, you use min.

It is usually perfectly polite in everyday Danish.

Kan du ... ? is a very common way to ask someone to do something:

• Kan du hjælpe mig? = Can you help me?
• Kan du ringe til hende? = Can you call her?

It can literally refer to ability, but in normal conversation it often works as a polite request, just like English Can you ... ?

If you want to sound softer, you could add words like:

• lige = just
• måske = maybe

For example:

• Kan du lige give beskeden videre til min søster?

Both can be used as requests, but the nuance is slightly different.

• Kan du ... ? = Can you ... ?
  This often focuses on whether the person is able or available to do it, but it is also a normal request.

• Vil du ... ? = Will you ... ? / Would you ... ?
  This often sounds more directly like a request about willingness.

In many everyday situations, both are natural.
For example:

• Kan du give beskeden videre til min søster?
• Vil du give beskeden videre til min søster?

The first may feel a little more like Are you able to?
The second may feel a little more like Would you be willing to?

Not always.

Besked is a useful word with a few related meanings, such as:

• message
• information
• word
• sometimes even instructions or notice, depending on context

Examples:

• Jeg fik en besked fra ham. = I got a message from him.
• Giv mig besked. = Let me know.
• Hun gav klar besked. = She gave a clear message / made things clear.

So in your sentence, beskeden is best understood as the message, but the word is broader than just one exact English translation.

Yes, it is a very common expression.

At give noget videre means:

• to pass something on
• to relay something
• to hand something along

It can be used for:

• information
• a message
• an object
• responsibility, depending on context

Examples:

• Kan du give det videre? = Can you pass it on?
• Jeg giver informationen videre. = I’ll pass the information on.
• Hun gav beskeden videre til chefen. = She passed the message on to the boss.

So this is a good chunk to learn as a unit: give noget videre.

In everyday speech, native speakers might still say exactly this, but they may also use slightly looser versions such as:

• Kan du give min søster besked?
• Kan du lige give beskeden videre til min søster?
• Kan du sige det videre til min søster?

These are close in meaning, but the original sentence is already natural and correct.

A very common shorter phrase is:

• give nogen besked = let someone know / give someone a message

So depending on context, Danes may choose a shorter version if the meaning is obvious.