Is grammar THE must thing which should be CORRECT for mail writing?

Its obvious that grammar is important, but that isn't restricted to mails. It applies to other forms of communication as well. Mails and letters are 'recorded' forms of communication. Hence, it is possible to re-read the same. Therefore, unlike the oral form, any mistake that comes to the notice of the reader would keep getting noticed time and again.
I think there has been little thought on the logic behind the question. Imagine one is speaking in one's mother tongue. And imagine one is making many grammatical mistakes. Would it be well received by people who barely know the person? The answer is obvious.
Dinesh Divekar has made good points. Please consider them as well.

While grammar is important - what is more important is comprehension and the ability to express what is meant and how well the idea is made clear. However, this somewhat requires a good understanding of language and words and how they are used together.
An interesting video on English language:

Dear,
Few of the companies are giving training to their employees about mail writing etiquette. HR denotes a face of the organization; therefore, while writing a mail we should not only be conscious but same time assertive as well.
Always re-check what you wrote and modify if require. Fortunately,we are in generations where ready-made software is in built which not only correct spellings automatically even grammar too.
Don't restrict yourself...practice makes man/woman perfect.
Happy writings !!
Regards,
Gajendra Verma

Dear Aankit,
So many good views have been expressed.
In my view importance of grammar cannot be understated at any time, in any context.
Remember that our habits are characterized by what we repeatedly do.
Once we do not stick to the rules of grammar, over a period of time,there will be slackness resulting in poor communication performances.
V.Raghunathan

Grammer is like the button or zip to your attire.THINK?

Of course, if you are writing to someone, he or she could only understand what you want to communicate if you have correctly written that.
https://indianaccounting.blogspot.com

This quote will sum up the importance of Grammar.
Your grammar is a reflection of your image. Good or bad, you have made an impression. And like all impressions, you are in total control.
Jeffrey Gitomer
So take care of the commas, spellings and punctuations while writing official correspondence.

Grammar in any language is akin to Mathematics in many ways. If you are correct, you can score full marks in both. Talking or writing of grammar, one has to be very attentive even while spelling the word grammar. Failure to do so, will result in a glaring mistake. This thread is no exception.
V.Raghunathan

Actually, it depends on the subject you are gonna talk about. It is always best to maintain high-level of English slang to showcase your professionalism. The usage is essential in only certain sections of the sentence and just avoid overusing it.

Would always recommend -Not to use slang in official correspondence.