Czytanie wieczorem pomaga mi odpocząć.

Because czytanie is a verbal noun, meaning reading as an activity or thing.

In this sentence, czytanie is the subject of the verb pomaga:

• Czytanie = reading
• pomaga = helps

Polish often uses verbal nouns like this where English uses a gerund:

• Pływanie jest zdrowe. = Swimming is healthy.
• Czytanie pomaga. = Reading helps.

Using czytać here would not sound natural in standard Polish.

It is a neuter singular noun formed from the verb czytać.

So even though it comes from a verb, it behaves like a noun:

• it has gender: neuter
• it has case forms
• it agrees with verbs like a singular noun

That is why the verb is pomaga and not plural pomagają.

Because the subject, czytanie, is grammatically singular.

Even though English reading may feel like an activity rather than a thing, in Polish it is treated as a singular noun. So:

• Czytanie pomaga = Reading helps

If the subject were plural, the verb would also be plural.

Wieczorem is the form used to mean in the evening.

It is the instrumental case of wieczór, but here it functions like an adverbial time expression. Polish often uses this pattern for times of day:

• rano = in the morning
• wieczorem = in the evening
• nocą = at night

So czytanie wieczorem means reading in the evening.

Not in this meaning.

To express in the evening, the normal natural form is wieczorem.

You can say:

• w ten wieczór = on that evening
• wieczorem = in the evening

So for a general habit or time expression, wieczorem is the right choice.

Because pomagać takes the dative case, and mi is the normal unstressed dative form of ja.

• mi = to me
• mnie = to me, but usually stressed, emphasized, or used after a preposition

So:

• To mi pomaga. = This helps me.
• Mnie to pomaga, nie jemu. = It helps me, not him.

In your sentence, there is no special emphasis, so mi is the most natural choice.

Because the verb pomagać works like to help someone in Polish, and the someone is put in the dative.

The pattern is:

• pomagać komuś = to help someone

Examples:

• Pomagam mamie. = I help my mother.
• Muzyka pomaga mi się skupić. = Music helps me concentrate.

So mi is there because I am the person being helped.

Because after pomagać komuś, Polish often uses an infinitive to say help someone do something.

Pattern:

• pomagać komuś + infinitive

So:

• pomaga mi odpocząć = helps me rest / helps me relax

Other examples:

• To pomaga mi zasnąć. = It helps me fall asleep.
• Ćwiczenia pomagają mi lepiej się czuć. = Exercise helps me feel better.

This is a question of aspect.

• odpocząć = perfective, meaning to get some rest, manage to rest, or reach a rested state
• odpoczywać = imperfective, meaning to be resting or to rest regularly/in general

In this sentence, odpocząć is very natural because the idea is that reading helps you successfully relax / get rested.

So:

• Czytanie wieczorem pomaga mi odpocząć = reading in the evening helps me relax / rest

If you used odpoczywać, it would sound less natural here, though in some contexts it may still be understandable.

No, Polish word order is fairly flexible, but changing it can change the focus.

Your sentence is natural:

• Czytanie wieczorem pomaga mi odpocząć.

Other possibilities:

• Wieczorem czytanie pomaga mi odpocząć.
  This puts more focus on in the evening.
• Czytanie pomaga mi odpocząć wieczorem.
  This can sound more like it helps me rest in the evening, so the adverb may attach more to resting than to reading.

So the original version is good if you want reading in the evening as one idea.

Yes. Both are possible, but they are structured differently.

• czytanie wieczorem = reading in the evening
• wieczorne czytanie = evening reading

The first uses an adverbial expression of time.
The second uses the adjective wieczorne.

Both are natural, but czytanie wieczorem pomaga mi odpocząć sounds very ordinary and conversational.

In theory, Polish word order can sometimes allow slight ambiguity, but in this sentence the most natural reading is:

• reading in the evening helps me rest

That is because wieczorem comes right after czytanie, so listeners will usually connect those two.

If you wanted in the evening to clearly describe resting, you would more likely place it later:

• Czytanie pomaga mi odpocząć wieczorem.

A rough guide is:

• od-po-chonch

A few key points:

• cz sounds like English ch in chop
• ą before ć is usually pronounced something like on or a nasalized vowel
• final ć is a soft ch-like sound

A more careful approximation is:

• ot-po-chońć

The exact sound is hard to represent in English spelling, but the main thing is that cz is ch, and ą is nasal.