Jeg læser teksten nøje, før jeg afleverer opgaven.

Because Danish usually makes a noun definite by adding the article to the end of the noun.

• en tekst = a text
• teksten = the text
• en opgave = an assignment
• opgaven = the assignment

Both tekst and opgave are common-gender nouns, so their singular definite ending is -en.

In the present tense, Danish verbs normally have the same form for all persons.

So:

• jeg læser = I read / I am reading
• du læser = you read
• han/hun læser = he/she reads

Unlike English, Danish does not change the present tense verb depending on the subject, apart from using the same -r form across the board.

Here, nøje is being used adverbially, meaning it describes how the text is being read.

So læser teksten nøje means something like:

• read the text carefully
• read the text closely
• read the text thoroughly

English often uses -ly for this kind of word, but Danish does not have one single equivalent ending for all adverbs.

The most natural word order here is:

verb + object + adverb

So:

• Jeg læser teksten nøje.

That is the normal, neutral way to say it.
Putting nøje before teksten would sound marked or unnatural in ordinary Danish.

Because før jeg afleverer opgaven is a subordinate clause.

In Danish, some writers use start comma, which means they put a comma before subordinate clauses:

• Jeg læser teksten nøje, før jeg afleverer opgaven.

Others omit that comma:

• Jeg læser teksten nøje før jeg afleverer opgaven.

Both can be seen, depending on the comma style being used.

Because før introduces a subordinate clause, and subordinate clauses in Danish usually have subject + finite verb order:

• før jeg afleverer opgaven

In a main clause, Danish often uses verb-second word order, but after words like før, fordi, hvis, at, and når, the clause is subordinate, so the subject normally comes before the verb.

Here, før is a conjunction, because it introduces a clause:

• før jeg afleverer opgaven

If før were followed by just a noun phrase, it would function more like a preposition:

• før afleveringen = before the hand-in / before the submission
• før timen = before class

So in this sentence, it is specifically linking two actions.

Danish often uses the present tense when the time relationship is already clear from the sentence.

So:

• Jeg læser teksten nøje, før jeg afleverer opgaven.

can refer to a regular habit or a future sequence, depending on context.

English does something similar:

• I read the text carefully before I hand in the assignment.

You do not need a separate future form here.

Yes, you often can.

• Jeg læser teksten nøje, før jeg afleverer opgaven.
• Jeg læser teksten nøje, inden jeg afleverer opgaven.

Both are natural.
A useful difference is that inden is purely temporal, while før is more general and very common. In this sentence, either one works well.

Because afleverer is a finite verb in a full clause with its own subject:

• jeg afleverer opgaven

The word at is used with the infinitive:

• at aflevere = to hand in

So:

• før jeg afleverer opgaven = before I hand in the assignment

not:

• før jeg at aflevere opgaven ❌

Yes, in this kind of sentence you normally do.

The second part is a full subordinate clause:

• før jeg afleverer opgaven

That clause needs its own subject, even though it is the same person as in the first clause.

English can sometimes shorten this kind of structure, for example before handing in the assignment, but Danish usually does not just drop the subject in the same way here.