Учитель дал им домашнее задание.

Дал is the past tense form of дать.

It also tells you that the subject is:

• singular
• masculine

That is why it matches учитель.

Past tense in Russian often shows gender:

• дал = masculine
• дала = feminine
• дало = neuter
• дали = plural

So this sentence is specifically talking about a male teacher.

Because им is the dative case form of они.

With дать/дал, the person who receives something is usually put in the dative:

• они = they
• им = to them

So:

• учитель = the giver
• им = the receivers
• домашнее задание = the thing given

Because Russian often uses case endings where English uses a preposition.

In English, you often say gave homework to them.
In Russian, the idea of to them is already built into им, so no extra word is needed.

That is very common with verbs like:

• дать = to give
• показать = to show
• сказать = to say/tell
• помочь = to help

They often take a dative object without a preposition.

It is in the accusative case, because it is the direct object of дал.

The teacher gave what?
→ домашнее задание

However, for an inanimate neuter singular noun, the accusative looks exactly like the nominative:

• nominative: домашнее задание
• accusative: домашнее задание

So the form does not change, even though the case function does.

Because the adjective and the noun must agree.

• задание is a neuter singular noun
• домашнее is an adjective modifying it, so it must also be neuter singular
• both are in the same case here as well

That is why you get:

• домашнее
  • задание

Compare:

• домашний урок = masculine
• домашняя работа = feminine
• домашнее задание = neuter

This is an aspect question.

• дать → perfective
• давать → imperfective

Дал from дать usually presents the action as a completed whole: the teacher gave the homework.

Давал from давать would suggest something like:

• repeated giving
• habitual action
• background description
• emphasis on the process rather than the completed result

In a simple sentence about one completed act, дал is the natural choice.

No. Russian word order is flexible.

This version is a very normal, neutral order:

• Учитель = subject
• дал = verb
• им = indirect object
• домашнее задание = direct object

You can change the order to change emphasis, for example:

• Им учитель дал домашнее задание.
  Emphasizes to them
• Домашнее задание учитель дал им.
  Emphasizes the homework

So word order in Russian often reflects focus or information structure, not just grammar.

In this sentence, yes, because of дал.

The noun учитель is masculine, and the past-tense verb agrees with it in masculine form:

• учитель дал

For a female teacher, standard Russian would usually say:

• учительница дала им домашнее задание

So the sentence as written points to a male teacher.

Because Russian does not have articles.

English distinguishes:

• a teacher
• the teacher

Russian usually leaves that to context:

• Учитель can mean a teacher or the teacher
• домашнее задание can mean homework, the homework, or a homework assignment, depending on context

So you learn to understand definiteness from the situation rather than from separate words like a or the.

Yes. It is a very standard and common expression.

Literally, it is something like home assignment, but in normal usage it simply means homework.

You will also see related phrases:

• делать домашнее задание = to do homework
• задать домашнее задание = to assign homework
• проверять домашнее задание = to check homework

So this is a phrase worth learning as a unit.

A few parts would change depending on gender and number.

If the teacher were female:

• Учительница дала им домашнее задание.

If there were only one student:

• Учитель дал ему домашнее задание. = to him
• Учитель дал ей домашнее задание. = to her

So the forms change like this:

• им = to them
• ему = to him
• ей = to her

And the past tense verb changes with the teacher:

• дал = masculine
• дала = feminine

The stress is:

• учи́тель
• дал
• им
• дома́шнее
• зада́ние

So you can read it as:

Учи́тель дал им дома́шнее зада́ние.

Stress is important in Russian because it is not always predictable, so it is good to learn words with their stress from the start.