Когда курсор снова начал двигаться, я спокойно дописала письмо.

Because Когда курсор снова начал двигаться is a subordinate clause introduced by когда (when).

In Russian, subordinate clauses are separated from the main clause with a comma:

• Когда курсор снова начал двигаться, я спокойно дописала письмо.
• Я спокойно дописала письмо, когда курсор снова начал двигаться.

So the comma is required no matter which clause comes first.

Russian past tense agrees with the subject in gender and number.

• курсор is masculine singular, so: начал
• я refers here to a female speaker, so: дописала

Compare:

• male speaker: я дописал письмо
• female speaker: я дописала письмо

This is very normal in Russian: past tense shows gender in the singular.

Начал двигаться means started to move. It focuses on the beginning of the movement.

• курсор снова двигался = the cursor was moving again / moved again
• курсор снова начал двигаться = the cursor started moving again

So this version is more precise if the cursor had been frozen or still, and then movement resumed.

Because двигаться means to move intransitively, that is, the thing itself is moving.

• курсор двигается = the cursor is moving

But двигать usually means to move something:

• двигать стол = to move the table

As for двинуться, it is perfective and often means to set off / begin moving. After verbs like начать, Russian normally uses an imperfective infinitive, so начал двигаться is the natural choice.

Yes. Снова means again, and in many contexts опять can also be used.

• снова = again
• опять = again

In this sentence, both are possible:

• Когда курсор снова начал двигаться...
• Когда курсор опять начал двигаться...

A small nuance:

• снова often sounds a bit more neutral or slightly more written
• опять can sometimes sound more conversational, and in some contexts it can carry an ugh, again feeling

Here снова is a smooth, neutral choice.

This is about aspect and the meaning of the prefix до-.

• писала письмо = was writing / wrote a letter (process, not necessarily completed)
• написала письмо = wrote a letter (completed result)
• дописала письмо = finished writing the letter, wrote the rest of it

So дописала suggests that the letter was already partly written, and then the speaker completed it.

That makes it a very natural choice in this sentence: the cursor started moving again, and then she finished the letter.

Here спокойно means calmly, without stress, without rushing, or peacefully.

So it describes the manner of finishing the letter.

It usually does not primarily mean quietly in the noise sense. If you want quietly as in without making noise, Russian more often uses тихо.

So here:

• спокойно дописала письмо = finished the letter calmly / without fuss

Because it is the direct object, so it is in the accusative case.

However, for many inanimate neuter nouns, the accusative singular looks exactly the same as the nominative singular.

• nominative: письмо
• accusative: письмо

So even though the case has changed grammatically, the form stays the same.

Курсор is in the nominative case because it is the subject of the subordinate clause:

• курсор начал двигаться = the cursor started moving

So:

• курсор = who/what started moving? → nominative subject

The word order is not fixed, but the given order is very natural.

Russian often puts the time/background information first:

• Когда курсор снова начал двигаться, я спокойно дописала письмо.

You could also say:

• Я спокойно дописала письмо, когда курсор снова начал двигаться.

That still works, but the focus shifts slightly toward the main action first.

As for я: it can sometimes be omitted in Russian, but here it is useful because Russian past tense does not show person clearly.

• дописала by itself means feminine singular past, but not specifically I
• it could be I, you, or she, depending on context

So in isolation, keeping я is the normal choice.

A natural stress pattern is:

Когда́ курсо́р сно́ва нача́л двига́ться, я споко́йно дописа́ла письмо́.

Useful word stresses:

• когда́
• курсо́р
• сно́ва
• нача́л
• двига́ться
• споко́йно
• дописа́ла
• письмо́