Learn Hardcore Japanese: hatuon wo kakuninsuru toki ha, heddohon wo site reibun wo nando mo kiite iru. - When I check pronunciation, I put on headphones and listen to the example sentences many times.

確認するとき = when I check (the checking and the main action happen about the same time; a general, repeated situation)
確認したとき = when I checked (after I had checked; more past/completed feel)

Here, this is a general routine: When I check pronunciation, I do X, so the dictionary form 確認するとき is used.

Also, using に (確認するときに) would be possible, but:

ときは focuses on that situation as a topic.
ときに focuses more on the time point when something happens.

In a habitual description, 〜ときは is very natural.

What is the function of は after とき in 確認するときは?

Here は is the topic marker attached to とき:

確認するときは = As for when I check pronunciation / When I’m checking pronunciation, …

This sets up that situation (the time when you check pronunciation) as the topic, then the rest of the sentence explains what you do in that situation.

If you said 確認するときに, you’d sound more like you’re focusing on the time point, not on describing your usual way of doing it. 〜ときは is common when explaining habits, rules, or typical behavior.

Who is the subject of the sentence? There is no I or you written.

Japanese frequently omits the subject when it is clear from context. In this sentence:

発音を確認するときは、ヘッドホンをして例文を何度も聞いている。

the unspoken subject is most naturally I:

(私は) 発音を確認するときは、ヘッドホンをして例文を何度も聞いている。
= When I check pronunciation, I put on headphones and listen to the example sentences many times.

Depending on context, it could also refer to you, we, etc., but by default, a statement about one’s own routine will usually be understood as I.

What exactly does ヘッドホンをして mean? Why する with ヘッドホン?

ヘッドホンをする is a common expression meaning to put on / wear headphones.

ヘッドホン = headphones
を = marks the object
する (in して) = to do, but with some items it takes on the meaning to wear / put on

Examples of similar patterns:

ネクタイをする = to wear a tie
指輪をする = to wear a ring

So ヘッドホンをして = (I) put on headphones and… / (with) headphones on, (I)…

Could I say ヘッドホンをつけて instead of ヘッドホンをして? Is there a difference?

Yes, ヘッドホンをつけて is also understandable and used, meaning to put on headphones.

Nuance:

ヘッドホンをして
- very common and natural
- emphasizes wearing them (state of having them on)
ヘッドホンをつけて
- slightly more focused on the action of attaching/putting on
- still fine in everyday speech

In this sentence, because it describes a habitual method (how you usually do it), ヘッドホンをして fits very well.

What does 例文 mean here? Is it just “example”?

例文 literally means example sentence:

例 = example
文 = sentence / text

In language-learning contexts, 例文 usually refers to sample sentences that illustrate how a word or grammar pattern is used. So here it means:

listen to the example sentences (probably from a textbook, app, or audio material).

What does 何度も mean, and what is the role of も in it?

何度も means many times / over and over.

何度 = how many times / how often
も here emphasizes a large or repeated number of times

So:

何度 (by itself, in a question) = how many times?
何度も (in a statement) = many times / repeatedly

The も adds an emphatic feeling: again and again, countless times, over and over.

What’s the difference between 何度も and 何回も?

Both 何度も and 何回も can mean many times in everyday speech, and they often overlap. Some nuances:

何度も
- Common in set phrases like 何度も言うけど (I’ve told you many times, but…)
- Slightly more general; often used for actions or experiences.
何回も
- Slightly more concrete, like counting occurrences (times, turns).
- You might use it when you are thinking of a specific number of attempts or tries.

In this sentence, either could work, but 何度も is very natural for listening many times.

Why is the verb 聞く in the form 聞いている at the end, instead of 聞く or 聞きます?

聞いている is the ている form of 聞く.

The 〜ている form has two main uses:

Ongoing action (I am currently doing X)
Habitual or repeated action / current state (I do X regularly / I am in a state of X)

Here, the whole sentence describes a habitual action / method:

聞いている = (I) listen (repeatedly / as a habit)

Using simple 聞く could sound more like a general statement without that ongoing/habitual nuance, and 聞きます would be the polite form. Since the sentence is in plain style, 聞いている fits:
When I check pronunciation, I (make a habit of) listening to the example sentences many times (with headphones on).

Is 聞いている here more “I am listening now” or “I (always) listen”?

In isolation, 聞いている could mean either:

I am listening (right now)
I listen (habitually / usually)

In this sentence, combined with 発音を確認するときは (when I check pronunciation), it clearly describes a typical routine or method, so it’s understood as:

I (normally) listen many times
not just a one-time “I am listening right now.”

Why is the sentence in plain form (聞いている) and not polite (聞いています)?

Japanese has two main levels here:

Plain: 聞いている
Polite: 聞いています

This sentence is entirely in plain form:

確認する (plain)
聞いている (plain)

Plain form is used:

In writing (diaries, notes, explanations to oneself)
Among friends / equals
In textbooks as example structures

If you wanted a polite version, you could say:

発音を確認するときは、ヘッドホンをして例文を何度も聞いています。

Could the word order be 例文を何度も聞いている instead of 何度も聞いている after 例文を?

In the original:

…ヘッドホンをして例文を何度も聞いている。

This already has 例文を immediately before 何度も聞いている. You could also reorder slightly:

例文を何度も聞いている (as it is)
何度も例文を聞いている

Both are grammatical and natural. The nuance is very small:

例文を何度も聞いている
- Slightly more neutral; 例文を聞く is the basic phrase, with 何度も modifying 聞く.
何度も例文を聞いている
- Puts a bit more emphasis upfront on many times.

In practice, both are fine in this context.

Is the comma 、 after は in ときは、ヘッドホンをして… necessary?

The 、 (Japanese comma) is not grammatically required, but it is standard and helpful.

発音を確認するときは、ヘッドホンをして…
The comma visually and rhythmically separates the condition/time clause (発音を確認するときは) from the main action (ヘッドホンをして…).

You could technically omit it:

発音を確認するときはヘッドホンをして例文を何度も聞いている。

but most writers would include the comma for readability.

Could you rewrite the whole sentence in a clearer, more explicitly “I”-focused English?

A natural, explicit translation would be:

When I check my pronunciation, I put on headphones and listen to the example sentences over and over.