First of all, there is no term as "asked to leave" under any labour law. Either an employee resigns voluntarily or the company terminates his service. Asking someone to resign is voilation of law and amounts to illegal termination.
Generally speaking the company has more liabilities/ burden in case of termination of an employee and that is the sole resign they illegally "ask to resign" and obtains resignation from the employee.
If someone is terminated on any misconduct then such misconduct has to be proved. If the company terminates due to decrease is sale/ profit etc. then it comes under retrenchment and in such case retrenchment compensation has to be paid depending on the tenure of service, wages etc. Simple notice period payment is irrelevant.
Now coming to your queries, the answers are in bold
1. Now my Decemeber salary is also due which is not part of full and final settlement, Can I send notice for non payment of wages. Can I also ask them to pay penalty/interest for non payment of wages? and normally how much time does it take to resolve such cases?
Surely you can send notice demanding the outstanding wages. You are also entitled to interest/ compensation for the delayed payment. Send the notice personally or through lawyer under registered/ speed post. Time for resolving the issue cannot be predicted.
2. Can HR send mail of acceptance of resignation even witohut receiving written resignation from me? Isn't this a case of termination where HR should clear all the dues on the last working day?
HR cannot send you acceptance of resignation when you have not sent any resignation. Since you have already received the acceptance of resignation, immediately reply in writing to the HR that you have not tendered any resignation. If you don't do it the company may take the plea that you have tendered the resignation 'verbally' and thus they have accepted it. You have to challenge the resingation ASAP if you will to proceed legally.
3. If i want to legally fight with the organisation, what will be the cost involved?
There will hardly be any charge in relation to court fees etc. But you have to bear the legal expenses of lawyer. Now the professional fees depend from one lawyer to another and also depends on the geographical location of the court. So it is better to consult a local lawyer in your area and find out the approximate expenses that you might have to incurr.