Learn Hardcore Japanese: watasi no kuni de mo nihongo wo benkyousuru hito ga imasu. - In my country too, there are people who study Japanese.

私の国で日本語を勉強する人がいます。
= In my country, there are people who carry out the action of studying Japanese there.
私の国に日本語を勉強する人がいます。
= In my country, there are people who exist who study Japanese. (More like “In my country, there are such people”.)

Both can be grammatically possible, but with 勉強する (an action), で is the natural default to express “the place where they study.”

Why is it 私の国でも with も attached to で (making でも), instead of something like 私の国もで?

も attaches to the particle, not the noun, when you want to say “also/too” about that whole phrase.

私の国でも literally: even / also in my country
Here, でも is で + も: “at/in + also”.

If you said 私の国もで, it would be ungrammatical: も must come after the word or particle it modifies. So your choices are:

私の国も … (my country also …)
私の国でも … (also in my country …)
私の国にも … (also in my country …, with に)

In this sentence, the point is “In my country too, there are people who study Japanese”, so でも is highlighting the location “in my country” as the thing that is “also” true, probably in contrast to some other place that was just mentioned.

What is the difference between 私の国でも日本語を勉強する人がいます。 and 私の国にも日本語を勉強する人がいます。?

Both are possible and close in meaning, but the nuance shifts slightly:

私の国でも日本語を勉強する人がいます。
- で emphasizes the place where the studying happens.
- Nuance: In my country too, people (there) study Japanese.
私の国にも日本語を勉強する人がいます。
- に emphasizes the place where such people exist.
- Nuance: In my country too, there exist people who study Japanese.

In many real-life contexts, they will be understood almost the same, and both are natural. Because 勉強する is clearly an action, many speakers will lean slightly toward で as the default, but に is not wrong.

Why is 日本語 marked with を here? Could it be 日本語が勉強する?

With 勉強する, the thing you study is normally marked with を as the direct object:

日本語を勉強する = to study Japanese

You cannot say 日本語が勉強する in this meaning, because that would literally mean “Japanese studies” (as if Japanese were the subject doing the studying). That makes no sense here.

So:

日本語を勉強する人
= people who study Japanese (Japanese is what they study)

Why does 日本語を勉強する come in front of 人? Why not after?

In Japanese, a phrase that describes a noun comes before the noun, similar to “people who study Japanese” in English, but with different word order:

日本語を勉強する人
Literally: [Japanese (を) study (する)] person
Meaning: a person who studies Japanese / people who study Japanese

This is called a relative clause.
You cannot move 人 in front of 日本語を勉強する in this pattern; putting 人 after the describing phrase is the basic rule:

✅ 日本語を勉強する人
❌ 人日本語を勉強する (ungrammatical as a noun phrase)

Why is it the plain form 勉強する and not 勉強しています or 勉強している before 人?

When you use a verb to modify a noun (make a relative clause), Japanese normally uses the plain (dictionary) form to express:

habitual actions
general characteristics
timeless facts

So:

日本語を勉強する人
= people who (typically / generally) study Japanese

If you said 日本語を勉強している人, that focuses more on the ongoing action: “people who are studying Japanese (now / currently / in progress)”. Both forms can be correct, but:

勉強する人 = people who study Japanese (as a regular or general thing)
勉強している人 = people who are in the state of studying Japanese (e.g., currently enrolled, in the middle of studying)

What does が do in 人がいます? What would change if it was 人はいます instead?

が marks the subject that is being introduced or described. In 人がいます, it states the existence of such people:

人がいます。 = There are people.

If you say 人はいます, は marks 人 as the topic, often with a contrastive nuance, like “as for people (who study Japanese), they do exist (though something else might not).” For example:

私の国では中国語を勉強する人はいませんが、日本語を勉強する人はいます。
(There aren’t people who study Chinese in my country, but as for people who study Japanese, they do exist.)

So in your original sentence, 人がいます is the neutral way to say “there are such people” without extra contrast.

Why do we use います instead of あります?

Japanese uses two different “there is/are” verbs:

います for living / animate things (people, animals, etc.)
あります for inanimate things (objects, events, plants, abstract things, etc.)

Because 人 (people) are animate beings, you must use いる / います:

人がいます。 = There are people.
本があります。 = There is a book.

So 人があります would be incorrect.

Can we change the word order in 私の国でも日本語を勉強する人がいます。? For example, can 日本語を勉強する人 come before 私の国でも?

Japanese word order is somewhat flexible, but some parts are more fixed than others.

You can move the location phrase around, for example:

日本語を勉強する人が私の国でもいます。
日本語を勉強する人は私の国でもいます。 (with a topic nuance on 人)

These are grammatically possible, but the default / most natural order is often:

[Location] で [Action phrase] がいます。
→ 私の国でも日本語を勉強する人がいます。

Moving 私の国でも to the middle or the end is usually for emphasis or contrast, and can sound more marked or “spoken-style” depending on context. The original word order is the most straightforward.

In this sentence, what exactly does も (“also/too”) refer to? Does it mean “In my country too,” “Japanese too,” or “people too”?

Here, も is attached to で (making でも), so it applies to the location phrase 私の国で:

私の国でも = In my country too

The implied contrast is something like:

In some other place(s), there are people who study Japanese, and
In my country too, there are people who study Japanese.

If you wanted to say “Japanese too” (as in: they study English and Japanese too), you’d put も on 日本語:

私の国で日本語も勉強する人がいます。
= In my country, there are people who also study Japanese (in addition to something else).

If you wanted “people too” (as in: there are students and there are also people who study Japanese), you’d put も on 人:

私の国で日本語を勉強する人もいます。
= In my country, there are also people who study Japanese.

So the position of も is crucial for what “also” attaches to.

Could I just say 私の国でも日本語を勉強します。 to mean the same thing?

私の国でも日本語を勉強します。 is grammatically fine, but the meaning is different:

私の国でも日本語を勉強します。
= (I / we) also study Japanese in my country.

Here, the subject is implied and usually understood as 私 (I) or 私たち (we). It’s about what you/your group do, not about the existence of people who do it.

The original sentence:

私の国でも日本語を勉強する人がいます。
= There are people in my country who study Japanese.

This talks about some people (unspecified), not necessarily including the speaker. So if the point is “some people in my country study Japanese,” you need the 人がいます pattern.