Learn Hardcore Danish: Studenten læser bogen på biblioteket. - The student reads the book at the library.

So studenten literally means student-the, and bogen means book-the.
That’s why you don’t see a separate word like the in front of them.

What are the base forms and genders of the nouns in this sentence?

The base (indefinite singular) forms are:

en student – common gender (like most people/thing words)
en bog – common gender
et bibliotek – neuter gender

Their definite singular forms are:

studenten (the student)
bogen (the book)
biblioteket (the library)

Common gender nouns take en in the indefinite and usually -en in the definite.
Neuter gender nouns take et in the indefinite and usually -et in the definite.

Can læser mean both reads and is reading?

Yes. Danish does not have a separate continuous form like English is reading.

Studenten læser bogen can mean:
- The student reads the book (habitually or generally), or
- The student is reading the book (right now)

Context normally makes it clear which meaning is intended.

How is the verb læse conjugated, and why doesn’t it change with the subject?

Danish verbs do not change form for person or number. The present tense is the same for I/you/he/she/we/they.

For at læse (to read):

jeg læser – I read / I am reading
du læser – you read
han/hun læser – he/she reads
vi læser – we read
I læser – you (plural) read
de læser – they read

Past tense is læste for all persons: jeg læste, han læste, vi læste, etc.
So læser stays the same in Studenten læser bogen, regardless of who the subject is.

Why is the preposition på used with biblioteket instead of i?

With many public places and institutions, Danish uses på where English uses at:

på biblioteket – at the library
på universitetet – at the university
på arbejde – at work
på hospitalet – at the hospital

På biblioteket focuses on the place as an institution or activity location (going to the library, spending time there).
I biblioteket literally means in the library, inside the physical building, and is used less commonly in this kind of neutral example, but can be correct if you want to stress the inside location.

Can I say Studenten læser på biblioteket bogen?

No, that word order is not natural Danish.

The neutral order in statements is:

Subject – Verb – Direct object – Other information (place, time, etc.)

So:

Studenten læser bogen på biblioteket. ✔

If you want to emphasize the place, you can put the place first:

På biblioteket læser studenten bogen. ✔ (emphasis on at the library)

But Studenten læser på biblioteket bogen is wrong; the direct object bogen should not be placed after the place phrase like that in a normal statement.

Is studenten the normal Danish way to say the student?

Grammatically, studenten does mean the student, from en student.

However, in modern Danish:

en student often means someone who has graduated from gymnasium (upper secondary school), not just any student.
For a current university/college student, en studerende is more common.

So in everyday language, you’ll often hear:

den studerende læser bogen på biblioteket – the (university) student reads the book at the library.

Your sentence with studenten is grammatically fine but a bit more limited in who it describes.

How would I say A student reads a book at the library?

Change the definite forms to indefinite:

En studerende læser en bog på biblioteket.
- This is the most natural way if you mean a university student.

Using the same noun as your example:

En student læser en bog på biblioteket.

Both are grammatically correct; en studerende is more typical in everyday Danish for a student at a university.

How do you pronounce læser, and what sound does æ have?

Læser is typically pronounced approximately like [ˈlɛːsɐ].

æ is similar to the vowel in English “bed”, but often a bit more open and here it’s long: lɛː.
The r at the end is weak; the final -er often sounds like a reduced -ə sound (-uh).
So you can think of læser roughly as “LEH-ser”, with a long eh sound and a soft ending.

(Exact pronunciation varies slightly by region, but this is a good standard guideline.)

Why are only the first word and not the nouns capitalized?

In Danish, nouns are not capitalized by default (unlike German).

The rules are similar to English:

Capitalize the first word of a sentence.
Capitalize proper nouns (names of people, cities, countries, etc.): Peter, København, Danmark.

So in Studenten læser bogen på biblioteket., only Studenten is capitalized because it is the first word in the sentence, not because it is a noun.

How do I make the sentence plural: The students read the books at the library?

You need plural forms and plural definite endings:

en student → studenter → studenterne (the students)
en bog → bøger → bøgerne (the books)

So:

Studenterne læser bøgerne på biblioteket.

A very common everyday variant using studerende would be:

De studerende læser bøgerne på biblioteket.
(literally the studying-ones read the books at the library; de studerende = the students)

Can læser also mean studies in the sense of studying at a university?

Yes. At læse can mean:

to read (a book, a text)
to study (a subject) at a university or similar

Examples:

Hun læser bogen. – She is reading the book.
Hun læser medicin. – She studies medicine.
Han læser på universitetet. – He studies at the university.

In your sentence Studenten læser bogen på biblioteket., læser is clearly reads / is reading, because it has a direct object bogen (the book).