Du burde tage din paraply med, hvis det regner.

Yes, historically burde is the past form of bør. But in sentences like this, it usually does not mean past time. It works like English should: a recommendation, advice, or a mild obligation.

So Du burde tage din paraply med means something like You should take your umbrella with you, not You should have taken...

Both can mean should / ought to, but they often feel a little different:

• Du bør tage din paraply med = a bit firmer, more direct
• Du burde tage din paraply med = often a little softer, more tentative

In everyday Danish, burde is very common for giving advice.

Because tage ... med means take/bring along.

• tage din paraply = take your umbrella
• tage din paraply med = take your umbrella with you / bring your umbrella along

In this sentence, med makes it clear that the umbrella should come with you.

Because tage med behaves like a verb + particle combination. In Danish, the object often goes between the verb and the particle:

• tage din paraply med
• tage den med

So even though tage med belongs together in meaning, it is often split in the sentence.

Yes. Very often.

Danish uses tage ... med in places where English would naturally say bring rather than take. So the best English translation may be:

• Bring your umbrella with you
• Take your umbrella with you

Both are fine depending on context.

Because with du, Danish uses din/dit/dine.

sin/sit/sine is the reflexive possessive that is mainly used with a third-person subject:

• Han tager sin paraply med = He takes his own umbrella
• Hun tager sin paraply med = She takes her own umbrella

But with du, you say:

• Du tager din paraply med

So din is correct here.

Because din already works as the determiner.

In Danish, you normally do not use an article together with a possessive:

• din paraply = your umbrella
• not en din paraply
• not den din paraply

Also, the basic noun is en paraply.

Here det is a dummy subject, just like English it in it is raining.

It does not refer to a thing. It is simply required by the grammar of weather expressions:

• det regner = it is raining
• det sner = it is snowing

So det is there because Danish, like English, usually wants a subject in the sentence.

Because Danish usually uses the present tense after hvis for future conditions.

So:

• hvis det regner = if it rains / if it is raining

This is very similar to English, where we also normally say if it rains, not if it will rain.

Because hvis means if, while når means when.

• hvis det regner = if it rains, maybe it will happen, maybe not
• når det regner = when it rains, implying it does happen or is expected to happen

So hvis is the right choice for an open condition.

Because hvis introduces a subordinate clause, and subordinate clauses in Danish normally keep subject + verb order:

• hvis det regner

There is no inversion inside that clause.

But if the hvis-clause comes first, the main clause shows inversion:

• Hvis det regner, burde du tage din paraply med.

Notice burde du instead of du burde.

You can, but the meaning becomes slightly less specific.

• Du burde tage din paraply. = You should take your umbrella.
• Du burde tage din paraply med. = You should take your umbrella with you.

In this kind of sentence, med sounds more natural because it clearly expresses bringing it along.