He does everyday all the sameis routine is the same every day - going to the university and then coming back home.

"everyday" (one word) is an adjective that means basically "usual" / "normal" / "routine"

If you want to say someone does something every day, that's two words.

He has three friends - they all like to hang out together but it's become rare lastin the last few months.

Jimmy iIn his free time absorbJimmy relaxes by playing the guitar, it clears his mind, makes him calm and opens the heart of creativity.

He want to make songs, not just singing covers, but he doesn't hasve members for his band.

His best friends don't really like playing music, they love another types of activities.

Making a (music / a song) alone - it's a challenge without motivation, joy and support.

music is uncountable so doesn't use "a".

He had started trying to find a new members, but accordingdue to his modesty and introversion, he gained nohadn't found success.

One day, Jimmy met with Kayla by chance and surprisingly they started to communicatetalk.

Communicate is too academic/abstract to fit here.

Jimmy played to her his demos to her and suggested Kayla to join makingshould join and they should make music together.

She rejected the idea at first cause she didn't know how to play onany music instruments.

Since then, Jimmy started making ahis dream come true.

WAlong with Kayla he found a drummer, Alexandra.

They called yourself "The Rackenboats", a weird name that Jimmy came up with.

Jimmy, Kayla and Alexandra hopes that one day someone will hear their music and his heart will be full of sincere emotions that music gives to people.

He does everyday all the same - going to thethe same thing everyday - go to university and then cominge back home.

(1) Alternatively: "For him, every day's the same - go to university and then come back home."
(2) In this context, we don't use "the" before "university", the same way we say "I'm going to school" but not "I'm going to my school". Instead, "the university" would be used when we want to place emphasis on the fact that it's a specific university in particular.

He has three friends - they all like to hang out together but it's become rare lastover the last few months.

Jimmy iIn his free time, Jimmy is absorbs byed in playing the guitar,; it clears his mind, makes him calm and opens the heart ofunlocks his creativity.

(1) I restructed your sentence slightly so it sounds more natural.
(2) Alternatively, instead of "makes him calm", you can say "calms him down".
(3) Perhaps in Russian you have the phrase "opens the heart of creativity", but in this context, I think it's a slightly questionable expression in English. I might suggest "unlocks his creativity" instead.

He wants to make songs, not just singing covers, but he doesn't hasve members for his band.

His best friends don't really like playing music, they love another types of activities.

Making a music alone - it's a challenge withouthen you don't have motivation, joy and support.

(1) "Music" is an uncountable noun.
(2) "Without" here isn't wrong, but I think my suggestion is clearer. Your original sentence might be easily interpreted to mean that making music is a challenge that has no motivation, joy or support.

He started trying to find a new members, but accordingdue to his modesty and introversion, he gainehad no success.

The grammatical number is inconsistent. "A" is singular, while "members" is plural.

One day, Jimmy met with Kayla by chance and surprisingly they started to communicatetalk.

(1) "Meet with" suggests that the meeting was planned beforehand.
(2) "Communicate" strikes me as too stiff and formal a word here.

Jimmy played to her his demos and suggested Kayla to jo(that) Kayla join him in making music together.

The phrase is "join (someone) in (doing something)".

She rejected him at first because she didn't know how to play on musical instruments.

SinceFrom then on, Jimmy started making a dream come true.

The three young musicians rehearsed their strange and heartbreaking songs every week.

They called yourselfthemselves "The Rackenboats", a weird name that Jimmy came up with.

Jimmy, Kayla and Alexandra hopes that one day, someone will hear their music and his heart will be full of sincere emotions that music givevokes toin people.

(1) "Jimmy, Kayla and Alexandra" are three people, so you should use "hope" (plural) instead of "hopes" (singular).
(2) Rewrote some parts of your sentence so that it sounds more fluent.