Jeg håber, at jeg får tid til at læse i aften.

No. They have two different jobs.

• The first at in Jeg håber, at jeg får tid ... means that. It introduces a subordinate clause.
• The second at in til at læse is the infinitive marker, like English to in to read.

So:

• håber, at jeg får tid = hope that I get time
• til at læse = to read

They look the same, but they do different things.

Yes, very often you can.

Both of these are natural:

• Jeg håber, at jeg får tid til at læse i aften.
• Jeg håber, jeg får tid til at læse i aften.

Leaving out at is common in everyday Danish. Including it can sound a little more explicit or slightly more formal, but both versions are standard.

That is about Danish comma style.

In Danish, many people write a comma before a subordinate clause, especially before at. So Jeg håber, at ... is very common.

You may also see:

• Jeg håber at jeg får tid til at læse i aften.

Both are accepted in modern Danish. So if you see the comma, it is not changing the meaning; it is just punctuation style.

Danish verbs do not change for person the way English verbs sometimes do.

So the present tense form stays the same with different subjects:

• jeg håber
• du håber
• han håber
• vi håber

And likewise:

• jeg får
• du får
• de får

In this sentence:

• håber = present tense of håbe
• får = present tense of få

Because Danish often uses the present tense for future meaning when the time is clear from the context.

Here, i aften already tells you that this is about later today. So jeg får tid can mean I get / I will have time.

This is similar to English in some cases:

• I hope I get time tonight.

So the present tense is completely normal here.

Literally, får tid means get time.

But in natural English, you would often translate it as:

• have time
• get time
• find time

So jeg får tid til at læse means something like:

• I have time to read
• I get time to read
• I find time to read

It is a very common Danish way to express the idea.

Here til links tid with the activity the time is for.

So:

• tid til noget = time for something
• tid til at gøre noget = time to do something

That is why Danish uses:

• tid til at læse
• literally: time for to read

Even though that literal wording sounds odd in English, it is normal in Danish.

Because after a subordinating conjunction like at, Danish normally uses subordinate clause word order.

That means the subject comes before the finite verb:

• at jeg får tid

In a main clause, Danish often follows verb-second word order. For example:

• I aften får jeg tid til at læse.

Here I aften comes first, so the verb får comes second, before jeg.

But after at, that inversion does not happen:

• ... at jeg får tid ...

So this sentence shows an important contrast between main-clause and subordinate-clause word order.

I aften means tonight or this evening.

It is a fixed time expression in Danish, so you do not use an article there. Danish often uses short adverbial expressions like this:

• i dag = today
• i morgen = tomorrow
• i aften = tonight / this evening

So i aften works as a complete time expression by itself.

Yes. The end position is the most neutral here, but it can be moved for emphasis.

Neutral:

• Jeg håber, at jeg får tid til at læse i aften.

More emphasis on tonight:

• I aften håber jeg, at jeg får tid til at læse.

Both are grammatical. The difference is mainly what you want to highlight.

It can mean both, depending on context.

• læse en bog = read a book
• læse til eksamen = study for an exam

In this sentence, with no object after it, læse will usually be understood as read in a general sense, unless the surrounding context is about school or exams.

So here most learners would naturally understand it as read.