No you don't have to root / unlock your phone to flag a stock firmware
6603 and 6653 are interchangeable... I used a 6603 firmware on my 6653 when MM was first released as I didn't want to wait for the AUS version...and was fine.. the only difference is that the d6603 has more LTE bands... So get the d6653 firmware for your carrier / country and use that and you will be fine.
Also , follow the link to the tutorial on the opening page of this thread for using flashtool.. that is what you will need to use to do this.
Also, a too, do a clean flash ( wipe your data ) ... Will give you best results.
Okay i will follow the above link,
One more thing that came into my mind the do i have to download the firmware according to my CDA version? My CDA version is 1288-5028-R9B
And also should i have to care about the "R9B" part of the cda version? Because my CDA version is only listed for UK region and that listed one's "Rxx" is also different from my phone's