What I have written is right?

We would say: The mirror is my best friend, because when I weep, it never laughs.”

No it’s not right.
You can have its mirror with no apostrophe, meaning the mirror of your special friend.
Or you can say of your special friend that it’s a mirror, meaning your friend is a mirror, or the mirror is your friend.
Assuming you meant the second thing, marinelife’s version would be correct.

The “mirror” sentence is fundamentally correct, other than the missing article “the”, as @marinelife pointed out. (I agree with your comma placement, and I think @marinelife added one that doesn’t belong—but reasonable writers can quibble about that.)

It’s your “topic sentence” that I take issue with. It’s not “incorrect” to ask “What I have written is right?”, but it’s an awkward-sounding question that identifies you as a non-English speaker. There are several ways that the question could be asked more normally:
“Is this right?”
“Have I written this correctly?”
“Is my writing correct?”
“Is what I have written correct?” (This is the closest to what you have, and I still wouldn’t use this version. Any of the three above would be preferable.)

I would ignore all of what @Zaku has advised. “Its mirror is…” makes no sense, and though you could say “A mirror is…” it wouldn’t read as well. You wouldn’t say “A mirror [indefinite] is my best friend”; you’d want to be specific, as @marinelife suggested: ”The mirror is…”)

@Zaku @marinelife :Thank you very much indeed!!
@CyanoticWasp :Special thanks to you too :)

@lessonenglish, here’s a suggestion that may help to make your questions clearer and our answers more helpful: make a definite separation between the sentence or sentences you are asking about and your explanation of them. That will focus our comments about the grammar just on the text you’re showing us instead of on the discussion.

Here’s an example of what I mean:

————————————————————————————-

My sentence

Mirror is my best friend because when I weep, it never laughs.

My question

Is my sentence grammatically right?

Explanation

Just asking this for confirmation, Someone has boyfriends, someone has girlfriends, someone has good friends but, I don’t have any of these. I have my special friend & it’s mirror.

————————————————————————————-

My own response is simply to add the article “The” before “mirror.” It doesn’t need another comma.

It’s a really nice poetic statement, if poetry is what you’re going for. I would change the pronoun to reflect your own sex. As in “Mirror is my best friend because when I weep, he (she) never laughs”.
That personifies mirror.
If that was your intention.

My two cents;
If you captialize Mirror, then you do not need “the”.
As in
“I have my special friend and it’s Mirror.”
That places mirror as if it is a name, not an object. Which fits with your poem.

Totally agree with @MissPoovey. Writing Mirror without “the” personfies it and makes the sentence more poetic. Personally, I see no need for the second comma. For me, it slightly slows the flow of the sentence.

I hasten to add that punctuation or lack of it—in my opinion—should be more about communication than adhering to a grammatical rule.

“Just asking this for confirmation, Someone has boyfriends, someone has girlfriends, someone has good friends but, I don’t have any of these. I have my special friend & it’s mirror. “Mirror is my best friend because when I weep, it never laughs”.

1) “Just asking this for confirmation” is a fragment. It would be better to start with a straightforward request: “Can you please tell me if this is grammatically correct?”

2) “Someone has boyfriends, someone has girlfriends” “Someone” usually refers to a specific but unknown single person. So you appear to be saying that one specific person has multiple boyfriends and girlfriends. That seems unlikely. You more likely intended to say “People can have boyfriends, girlfriends, or “best friends.”

3) “I have my special friend & it’s mirror.” This is odd construction. I would reword it “I have a special friend and it’s my mirror.”

4) “Mirror is my best friend because when I weep, it never laughs”. This should be personalized because you’ve identified it as your best friend. Therefore it’s not just any mirror, it’s YOUR mirror. In the USA, “weep” is not commonly used; cry is preferred. Also in the USA, the period always goes within the quotation marks. “My Mirror is my best friend because when I cry, it never laughs at me.”

mm, @GeorgeGee you’re right. If you were going for a poetic statement, it would read better as: “Some have boyfriends, some have girlfriends but I have Mirror.
Mirror never laughs when I weep.”

I would like to cast a dissenting vote on inserting the definite article, “The” before Mirror. While the sentence is more jarring the way the writer presented it, that is exactly the intent of the statement. Furthermore writing it as is makes if far more poetic. When asked for a best friend, most people would volunteer something like “Mary is my best friend.” or “Joe is my best friend.” By leaving out the word “The” you correctly capitalize the word mirror, thereby further showing its personification to the narcissist.

Now, the only word of caution if you actually believe this. Don’t fall in love with your best friend. The chapped lips you get from kissing Mirror really detract from the romance.

@lessonenglish, we are interpreting your question in different ways.

•   Some of us think you want comments on the accuracy of your entire “Details” section, and some think you are asking about just the one sentence in quotes.
•   Some of us think you are writing poetry and want an opinion on the quality of expression in your sentence, and some of us think you’re asking how this would be said in standard, conventional English.

Can you help us here by clarifying what it is you’re actually requesting?