После урока пианист снова сел за рояль и начал играть тише.

Because после requires the genitive case.

• урок = nominative singular
• урока = genitive singular

So после урока means after the lesson.

A useful comparison:

• после урока = after the lesson
• после уроков = after classes / after the lessons

Because пианист is the subject of the sentence, so it stands in the nominative case.

Here it is the person doing the actions:

• сел
• начал играть

So Russian uses the nominative: пианист.

Because сел means sat down or took a seat, while сидел means was sitting.

This is an aspect and meaning difference:

• сесть / сел = to sit down, to move into a sitting position
• сидеть / сидел = to be sitting

In this sentence, the pianist first sat down at the piano, so сел is the correct choice.

Literally, сел за рояль means something like sat down at the grand piano.

In Russian, за + accusative is often used with verbs of motion to mean moving into position at something, especially places where people sit and work:

• сесть за стол = sit down at the table
• сесть за компьютер = sit down at the computer
• сесть за рояль = sit down at the grand piano

So за here does not mean simply behind in the English spatial sense. It means taking one’s place at the instrument.

It actually is in the accusative case, but for a masculine inanimate noun, the accusative often looks exactly like the nominative.

So:

• nominative: рояль
• accusative: рояль

That is why the form does not visibly change here.

After начать / начал, Russian normally uses the infinitive.

So:

• начал играть = began to play
• not начал играл

This is just the standard pattern:

• начал читать = began to read
• начал работать = began to work
• начал играть = began to play

Because after начал, Russian usually uses an infinitive that names the action as an activity or process, and that is normally imperfective.

• играть = to play
• perfective would not sound natural here for the basic idea of starting an ongoing activity

The sentence is describing the beginning of the process of playing, not one completed act.

Тише is the comparative form of тихо and means more quietly, softer, or quieter.

So:

• тихо = quietly
• тише = more quietly / quieter

In this sentence, the idea is that he started playing more softly than before.

Russian often uses this comparative where English might say either:

• more quietly
• softer
• sometimes just quietly, depending on context

Usually yes, at least implicitly.

Here the sentence suggests that after sitting down again, the pianist began to play more quietly than earlier or more softly than expected.

Russian comparatives often work with an understood comparison, even when the second part is not stated explicitly.

Снова means again.

Here it modifies the action сел:

• пианист снова сел за рояль = the pianist sat down at the piano again

Russian word order is flexible, but this placement is very natural and neutral. Moving снова can change emphasis slightly, but the basic meaning stays the same.

No, Russian word order is fairly flexible. The given version sounds natural and neutral:

• После урока пианист снова сел за рояль и начал играть тише.

This order presents the information smoothly:

1. time: После урока
2. subject: пианист
3. action sequence: снова сел ... и начал играть тише

Other orders are possible, but they would usually change the emphasis rather than the core meaning.

Because Russian has no articles.

So урок, пианист, and рояль can mean:

• a lesson / the lesson
• a pianist / the pianist
• a grand piano / the grand piano

The exact meaning is understood from context. In this sentence, English naturally uses the in translation, but Russian does not need a separate word for it.