With that interior its got to be an '04-'06.
Mine's rally white outside and has the "blue" interior, but I liked the Velocity Red ones too. I still look when I see one on the streets.
That's the same KPU # that's on my '04, so it should work correctly, since they didn't change until '06.
An additional number is: IC:4238A-41846.
But as it is with the key, I believe you need one working remote to program another. That's likely why its not working for you. You may have to have...
What eric said. Tires make all the difference. Never underestimate them.... but it's not just about the money anymore. Spending money just gets you expensive tires that may be just as bad as the old one's. I've found that out the hard way.
Decide what YOU need for YOUR driving conditions. Decide...
You could also have the inside bore of the wheel centers flycut (re: machined) to 67.1mm, if there's room to safely do that? I once did it (by hand. NOT recommended) to a set of VW OE's I put on an early Civic.
I too have an NA Mazda3 with the TWM shifter, w/a heavyweight JBR knob, but this is not my first either.
I have modded shifters and linkage for years before this. It's a trade-off and only you can know whether its the right thing for you and your car.
Advantages: Shorter throws (obviously.)...