Эта надежда помогает мне спокойно ждать результата экзамена.

Because надежда is a feminine singular noun, and the demonstrative этот must agree with it in gender and number.

• этот = masculine
• эта = feminine
• это = neuter
• эти = plural

So:

• эта надежда = this hope

Because it is the subject of the sentence — the thing doing the action.

In this sentence:

• Эта надежда = the thing that helps
• помогает = helps

So надежда stays in the nominative singular.

The verb помогать / помочь normally takes the person being helped in the dative case.

So:

• я = I
• мне = to me

That is why Russian says:

• помогает мне = helps me / more literally helps to me

This is a very common pattern:

• Он помогает мне. = He helps me.
• Она помогает сестре. = She helps her sister.

Because спокойно is an adverb, and it describes how the waiting happens.

• спокойный / спокойная = calm, as an adjective describing a noun
• спокойно = calmly, as an adverb describing a verb

Here it modifies ждать:

• спокойно ждать = to wait calmly

So the sentence is talking about the manner of waiting, not describing a noun.

Because it depends on помогает.

The structure is:

• что-то помогает кому-то делать что-то
• something helps someone do something

So here:

• Эта надежда = this hope
• помогает мне = helps me
• спокойно ждать = to wait calmly

Using the infinitive after помогает is completely normal in Russian.

Because ждать traditionally takes the genitive case for the thing awaited.

So:

• результат = result
• результата = of the result / result (genitive form)

Therefore:

• ждать результата = to wait for the result

A useful thing to know: in modern Russian, especially in colloquial speech, ждать can sometimes also be used with the accusative, but the genitive is very standard and common, especially in more careful or neutral language.

This is a genitive chain.

First:

• ждать результата — результата is genitive because it depends on ждать

Then:

• результат экзамена = the result of the exam
• экзамена is also genitive because it depends on результат

So the whole phrase works like this:

• результата экзамена = of the exam result / more naturally for the exam result

In English we often use of or a noun-noun phrase, but Russian commonly uses the genitive for this relationship.

Помогает comes from помогать, the imperfective verb. It is used here because the sentence describes a general, ongoing, or repeated fact:

• This hope helps me...

It is not presenting a single completed act of helping. It is describing what this hope does for the speaker in general or at this moment.

The perfective partner is помочь, which would be used differently, for example in a completed event.

Because waiting is viewed here as an ongoing process, not a completed action.

Russian aspect matters a lot:

• ждать = to wait, to be in the process of waiting
• a perfective equivalent would express reaching some endpoint, but with wait Russian usually uses the imperfective when talking about the activity itself

Here the speaker means:

• to wait calmly as a process

That makes ждать the natural choice.

The given word order is very natural and neutral:

• Эта надежда помогает мне спокойно ждать результата экзамена.

Russian word order is more flexible than English because cases show the grammatical relationships. So other orders are possible for emphasis, for example:

• Мне эта надежда помогает спокойно ждать результата экзамена.
• Спокойно ждать результата экзамена мне помогает эта надежда.

These versions change the emphasis, not the basic meaning.

The original sentence is a good default, especially for learners.

No, it is natural if the hope has already been mentioned or is clear from context.

Russian uses этот / эта / это / эти much like English this:

• эта надежда = this hope

It points to a specific hope, not hope in general. If you wanted a more general idea, you might simply say надежда without эта, depending on context.

Yes. Adverbs in Russian are fairly flexible.

For example:

• Эта надежда помогает мне спокойно ждать результата экзамена.
• Эта надежда помогает мне ждать результата экзамена спокойно.

Both are understandable, but the first one sounds more natural and smooth in most contexts.

Putting спокойно before ждать is common because it clearly links the adverb with the infinitive.

A very useful way is this:

• Эта надежда = This hope
• помогает мне = helps me
• спокойно ждать = to wait calmly
• результата экзамена = for the exam result

So the structure is:

• This hope
  • helps me
    • wait calmly
      • for the exam result

That chunking makes the grammar much easier to see.