Yes, a private limited company with 12 or more employees is required to register for the Employees' State Insurance (ESI) and the Provident Fund (PF) schemes.
As for Professional Tax, it is a tax levied by state governments on individuals earning an income. The amount of tax and the threshold for tax liability vary from state to state. Employers are responsible for deducting and depositing professional tax on behalf of their employees to the state government. Therefore, you should check the professional tax rules in your state and ensure that your company is deducting and depositing the tax as required.