Урок заканчивается через десять минут.

Because урок is the subject of the sentence: it is the thing that is ending. The verb agrees with it in 3rd person singular:

• урок заканчивается = the lesson ends / is ending

The dictionary form is заканчиваться. This is the imperfective verb meaning to end or to come to an end.

Related aspect pair:

• заканчиваться — imperfective
• закончиться — perfective

So заканчивается is the 3rd person singular present-tense form of заканчиваться.

Here -ся helps make the verb intransitive: the lesson is not ending something else; it is itself coming to an end.

Compare:

• Учитель заканчивает урок. = The teacher ends the lesson.
• Урок заканчивается. = The lesson ends / is ending.

So:

• заканчивать = to finish/end something
• заканчиваться = to end

Russian often uses the present tense, especially the imperfective, for scheduled or expected near-future events. English does something similar in sentences like The class ends in ten minutes.

So Урок заканчивается через десять минут is a natural way to talk about something that will happen soon.

Yes, that is also correct.

The difference is mainly aspect and viewpoint:

• Урок заканчивается через десять минут — present-tense imperfective; often sounds like a scheduled or already unfolding situation
• Урок закончится через десять минут — future-tense perfective; focuses more directly on the completed endpoint

Both are natural, but заканчивается can feel a little more like the lesson is due to end in ten minutes.

With time expressions, через means in or after a period of time counted from now.

Examples:

• через минуту = in a minute
• через десять минут = in ten minutes
• через час = in an hour

In other contexts, через can also mean through or across, but here it has the time meaning.

In this time meaning, через requires the accusative case.

That is why the phrase is built as через десять минут, not with a genitive or dative form after через.

Russian number phrases follow special patterns.

With 1, the noun is singular:

• одна минута

With 2, 3, 4, the noun usually has the form:

• две минуты
• три минуты
• четыре минуты

With 5 and higher, the noun is usually in the genitive plural:

• пять минут
• десять минут
• двадцать минут

So минут is the correct form after десять.

Because десяти is not the form needed after через here. The preposition через requires the accusative, and the numeral десять has the form десять in the accusative.

So the correct phrase is:

• через десять минут

They mean different things:

• через десять минут = in ten minutes / ten minutes from now
• за десять минут = in ten minutes in the sense of within a duration of ten minutes

Compare:

• Урок заканчивается через десять минут. = The lesson ends ten minutes from now.
• Я сделал это за десять минут. = I did it in ten minutes.

This is a very important difference in Russian.

Yes. Russian word order is flexible, though not completely free.

For example, you could also say:

• Через десять минут урок заканчивается.

That puts more emphasis on the time expression. The original order, Урок заканчивается через десять минут, is a neutral and very natural way to say it.

Sometimes, yes, but they are not always exactly the same.

• урок usually means a school lesson or one class period
• занятие is broader and can mean a class, session, or lesson of many kinds

So if you are talking about a school timetable, урок is especially natural. If you mean a class session more generally, занятие may also work.