Я оставил заявку на курс русского языка.

Yes, оставить often means to leave something somewhere, but in some contexts it also means to leave behind / submit / provide.

In оставил заявку, the idea is I left/submitted an application. This is a normal Russian way to say that someone filled in or sent in a request/application, especially online or by phone.

So here оставил заявку is best understood as submitted an application / left a request.

That is the marker of the past tense in Russian.

The infinitive is оставить.
To form the past tense, Russian usually replaces -ть with -л:

• оставить → оставил = left / submitted
• делать → делал = did / was doing

In this sentence, оставил means submitted / left in the past.

Because Russian past tense agrees with the speaker's gender in the singular.

• оставил = a male speaker
• оставила = a female speaker

So:

• Я оставил заявку... = I submitted an application... said by a man
• Я оставила заявку... = the same thing, said by a woman

This is something English does not do, so it often stands out to learners.

Because it is the direct object of the verb, so it takes the accusative case.

The dictionary form is:

• заявка = application, request

But after a transitive verb like оставить, it changes:

• оставить заявку = to submit/leave an application

For feminine nouns ending in -а, the accusative singular often changes -а to -у:

• книга → книгу
• машина → машину
• заявка → заявку

Not exactly.

Заявка usually means a request, application, or submission, often in practical contexts such as:

• signing up for something
• sending a request online
• placing an order/request for service

Заявление is usually more formal and often means a written statement or official application/document.

So in this sentence, заявка is natural for signing up for a course. It sounds like the kind of request you leave on a website or with a school.

Because Russian uses на + accusative with many nouns for signing up for, buying a ticket for, or applying for an event/course/activity.

So:

• заявка на курс = application for a course
• билет на концерт = ticket for a concert
• записаться на курсы = enroll in courses

Here на курс means for a course.

Also, курс is masculine, and its accusative singular is the same as the nominative singular because it is inanimate:

• курс → на курс

Because after курс Russian often uses the pattern курс + genitive, meaning a course of ... or more naturally a course in ....

The basic phrase is:

• русский язык = Russian language

But after курс it becomes genitive:

• курс русского языка = course of the Russian language = Russian language course

Both words change because both belong to the noun phrase:

• русский → русского
• язык → языка

This is normal adjective-noun agreement in the genitive case.

Russian can sometimes shorten phrases in context, but курс русского языка is the full and clear expression.

• русского языка literally means of the Russian language
• this is the standard formal way to say Russian language course

You may hear shorter versions such as курс русского in conversation, where языка is understood. But the full version is more complete and neutral, especially in writing.

No, Russian word order is fairly flexible, although this version is the most neutral:

• Я оставил заявку на курс русского языка.

This sounds like a straightforward statement: I submitted an application for a Russian course.

Other orders are possible, for example:

• Заявку на курс русского языка я оставил.
• На курс русского языка я оставил заявку.

These alternatives may shift the emphasis, but the core meaning stays the same because the case endings show the grammatical roles.

Sometimes, but not always.

Russian often drops subject pronouns when the meaning is clear from context. However, in the past tense, the verb does not show person, only gender and number:

• оставил = masculine singular past
• but it does not by itself say I, you, or he

So Оставил заявку на курс русского языка could mean:

• I submitted an application...
• you submitted an application...
• he submitted an application...

Because of that, Я is often kept if the subject is not already obvious from context.

Оставить is the perfective verb, and оставлять is the imperfective partner.

Here оставил is used because the action is seen as completed: the application was submitted.

• Я оставил заявку... = I submitted/left an application. The action is done.

If you said Я оставлял заявку..., it would usually mean something more like:

• I was submitting / I did submit at some point / I had left an application

The imperfective version is used when the completion is not the main point, or when you are talking about the process, repetition, or experience. In this sentence, perfective is the natural choice because the result matters: the application has been sent.

A natural stress pattern is:

• Я оста́вил зая́вку на курс ру́сского языка́.

Word stress:

• оста́вил
• зая́вку
• ку́рс
• ру́сского
• языка́

Stress is important in Russian because it is not always predictable from spelling, and wrong stress can make a word sound unnatural.

Yes, literally it is of the Russian language, because it is in the genitive case.

Russian often uses this structure where English uses an adjective before a noun:

• курс русского языка = Russian language course
• literally: course of the Russian language

So the literal grammar and the natural English translation are a little different, but this is a very standard Russian construction.