Det begyndte pludseligt at regne.

In this sentence, det is a dummy subject. It does not refer to a specific thing.

Danish often uses det in weather expressions and other impersonal statements, just like English does in sentences such as It is raining or It started to rain.

So in Det begyndte pludseligt at regne, det does not mean a concrete it. It is simply required to fill the subject position.

Begyndte is the past tense of begynde, which means to begin or to start.

• begynde = to begin
• begynder = begins / is beginning
• begyndte = began

So Det begyndte pludseligt at regne is in the past and means that the rain started at a particular moment.

After begynde in this structure, Danish uses the infinitive with at:

• begynde at regne = to begin to rain

So:

• begyndte at regne = began to rain

You use regne, not regnede, because regne is the infinitive form. The tense is already shown by begyndte.

Here, at is the infinitive marker, like English to in to rain.

So:

• regne = rain / to rain
• at regne = to rain

After verbs like begynde, Danish often uses at + infinitive:

• begynde at læse = begin to read
• begynde at spise = begin to eat
• begynde at regne = begin to rain

Pludselig is basically the adjective form, meaning sudden.
Pludseligt is the form commonly used as an adverb, meaning suddenly.

So:

• en pludselig lyd = a sudden sound
• det skete pludseligt = it happened suddenly

In many cases, Danish adverbs are formed from adjectives by adding -t, especially when describing how something happens.

In Det begyndte pludseligt at regne, pludseligt comes after the finite verb begyndte and before the infinitive phrase at regne. That is a very natural position.

You may also see other word orders depending on emphasis, for example:

• Pludseligt begyndte det at regne.

This also means Suddenly, it began to rain, but now pludseligt is placed first for emphasis. Because Danish is a verb-second language, the verb begyndte must then come before det.

It is understandable, but it is usually less natural than Det begyndte pludseligt at regne.

Placing pludseligt before at regne sounds smoother and more idiomatic in this sentence. If you move pludseligt to the end, it can sound as if you are especially emphasizing the manner or timing of the raining itself.

So the most natural version for a learner to use is:

• Det begyndte pludseligt at regne.

They are similar, but begyndte at regne is the more standard and idiomatic choice.

• Det begyndte at regne = It began to rain
• Det startede at regne = understandable, but less natural to many speakers

In Danish, begynde at is very commonly used with actions like raining, speaking, laughing, and so on.

Yes, but the meaning is different.

• Det regnede = It was raining / It rained
• Det begyndte at regne = It began to rain / It started raining

So begyndte at regne focuses on the start of the rain, not just the fact that rain was happening.

Because Danish follows the verb-second rule in main clauses.

That means the finite verb usually comes in the second position. If you put something other than the subject first, the verb still stays second:

• Det begyndte pludseligt at regne.
• Pludseligt begyndte det at regne.

In the second sentence, pludseligt takes the first position, so begyndte must come next, and det moves after the verb.

The present-tense version would be:

• Det begynder pludseligt at regne.

That means It suddenly begins to rain or It suddenly starts to rain.

Compare:

• Det begynder pludseligt at regne. = present
• Det begyndte pludseligt at regne. = past

Yes, it is grammatical and natural. However, in everyday speech, many Danes would more commonly say:

• Det begyndte pludselig at regne.
• Pludselig begyndte det at regne.

You may notice that in real usage, pludselig is often used where learners might expect pludseligt. Both occur, but pludselig is very common in modern Danish, especially in speech.

So the sentence you have is correct, but it is useful to recognize both pludselig and pludseligt in real Danish.