In every year you are submitting your Annual Return of Income tax in any of the I.Tax office in the country and in that return form you should write your present address, that's all ,I do not think any separate formalities for address changes.
YES, this is necessary to inform along with duly filled application form, address proof and application fee Rs.94.00.
No doubt PAN is Permanent Account Number which is valid for whole life entire country but changes in address or city will change the assessing officer of your income tax. As you have moved from Chennai to Mumbai and the PAN card has issued in your old Chennai and your income tax assessing office is Chennai. In that case you have filed your return at Chennai. That's why you have to request for change of address in your PAN data.
You can also apply online for change of address in existing PAN by paying online payment and they give corrected card within a week.
PAN is valid all over India.
You don't have to file change of address.
You can furnish your tax return in the current state and city if you have address in the current residing place.
I had pan card of Chennai which I have been using for filing tax return in Trivandrum where I have got settled. I have not opted for change of address because I have own house in Chennai.
If you also have own house in previous place then you don't have to apply for change of address.
If not so, then it will be better for applying the change of address because if the tax authorities will be communicating only for the original address for any related correspondence. If you don't have any person responsible in forwarding the letter of tax authorities to you then you may not do the nodded formalities in time and that will end up as if you are following the rule. Every letter of tax authorities will carry some time limit.
So there is no hard and fast rule that you have to apply for change of address.