Is it your right to get a copy of FIR?
Yes! Its your right to get a copy of your FIR free of cost from the police

Should FIR be lodged ASAP? What will happen if there is a delay in filing FIR?
Yes, FIR must be lodged as soon as possible. If there is a delay in filing the FIR, it will help the accused because the accused shall get time to destroy the evidence, there shall be a problem in locating the witnesses and during the trial, the defense lawyer will make an allegation that FIR is an after thought.

Why they refuse to file FIR?
There are few reasons the police refuses to lodge FIR like the offense is not a cognizable offense. Sometimes they want to keep the crime graph low so if they don't lodge FIR, it will not show in the crime statistics in the area. Many a times, a party bribes police officers and that's why they refuse to lodge FIR. In most of mobile theft cases, it is a common practice that police does not register a FIR but registers a NCR so the graph of theft remains low.

What should you do if the police officer refuses to lodge a FIR?
If the police refuses to lodge FIR, don't think that this is the end of the road. The Criminal Procedure Code has a provision under section 156(3) where an aggrieved person may file a petition before the magistrate asking for a direction to the police to lodge a FIR.