Jeg er meget træt, men jeg læser alligevel et afsnit i bogen.

Because Danish, like English, normally uses to be with adjectives describing a state.

• Jeg er træt = I am tired
• er is the present tense of at være = to be

You would not use har here. Jeg har træt is not correct Danish.

Meget means very here, so meget træt = very tired.

Danish usually uses:

• meget with adjectives and uncountable ideas
• mange with countable plural nouns

So:

• meget træt = very tired
• mange bøger = many books

Here, træt is the basic singular/common form used after jeg er.

Danish adjectives can change form depending on grammar, but after jeg er, you normally use the basic form:

• Jeg er træt = I am tired
• Han er træt = He is tired

You get trætte in other contexts, for example:

• Vi er trætte = We are tired
• de trætte børn = the tired children

Because men means but, and it joins two main clauses:

• Jeg er meget træt
• jeg læser alligevel et afsnit i bogen

In standard Danish writing, a comma is normally used before men.

Because the second part is a full clause with its own subject and verb:

• jeg læser

In Danish, when two main clauses are linked by men, you normally state the subject again unless something is clearly omitted in a very different structure. So this sentence naturally keeps jeg in both clauses.

Alligevel means something like nevertheless, even so, or anyway.

It shows contrast: the speaker is tired, but despite that, the action still happens.

So:

• Jeg er meget træt, men jeg læser alligevel... = I am very tired, but I’m reading ... anyway / nevertheless

It is a very common word in Danish for this kind of contrast.

Because in a Danish main clause, the finite verb usually comes in the second position. This is the well-known V2 word order.

In the clause:

• jeg = subject
• læser = finite verb
• alligevel = adverb

So the natural order is:

• jeg læser alligevel et afsnit i bogen

If you move alligevel to the front, then the verb must still stay in second position:

• Alligevel læser jeg et afsnit i bogen

You cannot say Alligevel jeg læser... in standard Danish main-clause word order.

Because afsnit is a neuter noun in Danish, and neuter nouns take et in the indefinite singular.

• et afsnit = a section / a paragraph / a passage
• afsnittet = the section / the paragraph

Danish nouns are mainly either:

• en-words (common gender)
• et-words (neuter)

You simply have to learn which article each noun takes.

Afsnit can mean several related things depending on context:

• paragraph
• section
• passage
• sometimes a part of a text

In this sentence, et afsnit i bogen most naturally means a section or a passage in the book, though in some contexts it could also be understood as a paragraph.

It is not usually the normal word for a full chapter. That would be kapitel.

I usually means in.

So i bogen literally means in the book. Danish often uses i where English might also say in when talking about something contained within a text.

Examples:

• et ord i bogen = a word in the book
• et afsnit i bogen = a section in the book

Depending on context, English might sometimes prefer from the book, but Danish here naturally uses i because the section is part of the book’s contents.

Because Danish often puts the definite article at the end of the noun as a suffix.

So:

• bog = book
• bogen = the book

This is one of the big differences from English. Instead of a separate word like the, Danish often adds the ending directly to the noun.

Compare:

• en bog = a book
• bogen = the book

Læser is the present tense of at læse = to read.

In this sentence, it most naturally means I read / I am reading. But like the English present tense, Danish present tense can sometimes refer to the near future if the context makes that clear.

So jeg læser can mean:

• I read
• I am reading
• sometimes I will read

Here, the most natural interpretation is that the speaker is reading, or is going to read right now despite being tired.

Yes. If alligevel comes first in a main clause, the verb must still stay in second position.

So you can say:

• Jeg er meget træt, men jeg læser alligevel et afsnit i bogen.

Or:

• Jeg er meget træt, men alligevel læser jeg et afsnit i bogen.

Notice the difference:

• jeg læser alligevel
• alligevel læser jeg

This is a very important Danish word-order pattern.