P0303 is misfire on cylinder 3, since you say the plug and coil packs are good it is likely a dirty or faulty fuel injector.
You could try swapping the cylinder 3 fuel injector to a different cylinder and if the code follows, it will tell you the fuel injector was the issue.
No you need to have the key fob to start the vehicle, the auxiliary key just unlocks the door.
The auxiliary key is just used for when the key fob battery goes dead and you can't unlock the doors.
You hold the key fob next to the start button to start the vehicle.
Yes most likely the neutral safety switch, you can get an aftermarket switch online for around $20-$50.
https://www.carparts.com/details/Mazda/3/Replacement/Neutral_Safety_Switch/2008/i/4_Cyl_2-dot-0L/RM50640002.html?showfitment=true
Should be located just above the transmission oil pan.
go to the crutchfield website they will show you exactly what you need
https://www.crutchfield.com/car/carselector.aspx?lp=%2fcar%2foutfitmycar%2fmycar.aspx
by the way, you posted this in the wrong place