As per your details, it is clear that you are from Bangalore, which is in Karnataka State. The Shops and Establishment Act of Karnataka State is totally applicable to you. I have attached herewith the screenshot of Section 6A of the Karnataka Shops and Establishment Act, which makes it mandatory to provide a written appointment letter to an employee within 30 days. So, this is the first thing you are doing illegally.
Leave Entitlement Under the Karnataka Shops and Establishment Act
Another issue is Leave: Under Section 15(1) of the same act, every employee is entitled to leave of 1 day after every 20 days. This is over and above the weekly off, and for the purpose of counting 20 days, weekly offs and public holidays shall be counted. So, as per the act, giving at least 1.5 leave per month is mandatory, which you are not providing to your employee, which is illegal.
Under Section 15(2), an employee is entitled to 12 sick leaves, which again you are not providing to your employee, which is illegal.
Moreover, they are entitled to accumulate the leaves. My suggestion to you would be to mitigate this situation by issuing him an appointment letter backdated and providing him with the benefits of leaves as per the law above. However, I cannot advise on what to do regarding PF and ESI. Although I am unsure whether you are covered under the act or not. If covered, it could pose a problem for you if that employee approaches the authorities.
HR Responsibility and Compliance
As an HR person, it is your duty not to encourage these types of activities in your organization. Please apprise the management about the gravity of the situation and hope for the best.
Feel free to revert in case further clarification is required.
Regards.