Gra na gitarze pomaga mi odpocząć po pracy.

Here gra is a noun, not a conjugated verb.

So Gra na gitarze means playing the guitar or more literally playing on the guitar. This whole phrase is the subject of the sentence:

• Gra na gitarze = playing the guitar
• pomaga = helps
• mi odpocząć = me relax
• po pracy = after work

Even though gra can also be a form related to the verb grać in other contexts, here it functions as a noun meaning playing.

Yes. That is also correct, and many learners find it easier to understand.

• gra na gitarze = guitar playing / playing the guitar
• granie na gitarze = the act of playing the guitar

Both work, but they feel slightly different in style:

• Gra na gitarze pomaga mi odpocząć po pracy sounds very natural and idiomatic.
• Granie na gitarze pomaga mi odpocząć po pracy is also natural, and it makes the action sense a bit clearer.

Because Polish normally says grać na + instrument for many instruments.

So:

• grać na gitarze = to play the guitar
• grać na pianinie = to play the piano
• grać na skrzypcach = to play the violin

After na in this expression, gitara changes to gitarze. This is the locative case.

So:

• base form: gitara
• after na: na gitarze

Because the subject is singular.

The subject is Gra na gitarze, and that whole phrase is treated as one singular idea: playing the guitar.

So the verb must also be singular:

• Gra na gitarze pomaga... = Playing the guitar helps...

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

Mi means to me.

It is the short dative form of ja. With pomagać, Polish uses the dative:

• pomagać komu? = to help whom?

So:

• pomaga mi = helps me
• pomaga ci = helps you
• pomaga mu = helps him

In this sentence, mi odpocząć means helps me relax.

Both can mean me, but mi is the normal short, unstressed form here.

• pomaga mi = neutral, natural
• pomaga mnie = incorrect here
• pomaga mnie odpocząć = incorrect

If you wanted emphasis, you could use mnie, but usually in a different position and structure, for example:

• Mnie gra na gitarze pomaga najbardziej. = Playing the guitar helps me the most.

In your sentence, mi is the standard choice.

This is about aspect.

• odpocząć = perfective, to get some rest / to relax successfully
• odpoczywać = imperfective, to be resting / to rest regularly

In pomaga mi odpocząć, the idea is that playing the guitar helps you achieve relaxation after work. That is why odpocząć sounds very natural.

Compare:

• pomaga mi odpocząć = helps me relax / unwind
• pomaga mi odpoczywać = helps me be resting, helps me rest as an ongoing activity

The perfective form is usually the more natural choice in this exact sentence.

After po meaning after, Polish uses the locative case.

So:

• praca = work
• po pracy = after work

This same pattern appears in many time expressions:

• po obiedzie = after dinner
• po szkole = after school
• po spotkaniu = after the meeting

So pracy here is the locative singular form of praca.

The person in mi is the one who relaxes.

So the sentence means:

• Playing the guitar helps me relax after work.

Polish does not need to repeat the subject of odpocząć. It is understood from the dative pronoun mi.

Yes, Polish word order is fairly flexible.

The original sentence is a very neutral, natural order:

• Gra na gitarze pomaga mi odpocząć po pracy.

You could also hear variations like:

• Po pracy gra na gitarze pomaga mi odpocząć.
• Gra na gitarze po pracy pomaga mi odpocząć.
• Gra na gitarze pomaga mi po pracy odpocząć.

But the original version is probably the best one for a learner to use first.

Yes, gra can mean game, but it can also mean playing or performance, depending on context.

For example:

• to jest dobra gra = this is a good game
• gra na gitarze = playing the guitar

In your sentence, na gitarze makes the meaning clear, so nobody will think it means a literal game.

Yes, but the meaning changes slightly.

• Gra na gitarze pomaga mi odpocząć po pracy. = Playing the guitar helps me relax after work.
• Gra na gitarze pomaga odpocząć po pracy. = Playing the guitar helps one relax after work / helps with relaxing after work.

Without mi, the sentence becomes more general and less personal. With mi, it clearly describes your own experience.