Here það is a dummy subject, just like it in English weather expressions.
Examples:
Just as English says it rains even though it does not refer to a real thing, Icelandic uses það in the same way.
So in this sentence, það does not point to a specific object; it is simply required by the grammar of the weather expression.
Rignir is the 3rd person singular present tense of the verb að rigna (to rain).
So:
Because the subject is það (it), the verb is in the 3rd person singular form.
This is because Icelandic usually follows a verb-second (V2) word-order rule in main clauses.
The first element in the sentence is the subordinate clause:
After that, the finite verb of the main clause comes next:
Then the subject:
So:
This is similar to how some English structures invert after a fronted phrase, though Icelandic does this much more regularly.
A simpler comparison:
When something else comes first, the verb typically comes before the subject.
In standard Icelandic, that would usually be considered incorrect or at least nonstandard, because it breaks the normal verb-second pattern.
The standard form is:
So after the opening clause, the verb tek should come before ég.
Because regnjakka is the accusative singular form, used here as the direct object of tek (take).
The noun is:
But after tek in this sentence, it is the thing being taken, so it appears in the accusative:
So:
Með mér means with me.
The preposition með often takes the dative case, and the dative form of ég (I) is mér.
So:
Since með is followed here by the dative, you get:
In the full sentence:
Yes, very often taka ... með sér means to take ... along / to take ... with oneself.
Examples:
Notice that the pronoun changes depending on the subject:
So in your sentence, tek ég regnjakka með mér is a very natural way to say I take a rain jacket with me.
It can often cover both ideas, depending on context.
Icelandic present tense is commonly used for:
So this sentence could mean something like:
The exact nuance depends on the situation.
Either can work, depending on context.
So the most natural translation of the whole sentence is:
But grammatically, the Icelandic present tense itself is not limited to only one English tense.
Yes.
Hvort by itself often introduces an indirect whether question:
Hvort sem is more like whether ... or not / regardless of whether:
So hvort sem is especially used when the result is the same no matter which possibility is true.
A rough English-friendly guide would be something like:
A few useful notes:
This is only an approximation, but it can help you get started.
Yes, several useful things:
Weather expressions use a dummy subject
Icelandic often uses verb-second word order
Objects change form by case
Prepositions affect pronoun forms
Present tense can be flexible in meaning
So although the sentence is short, it shows several very central features of Icelandic grammar.