The loose spare tire sound for me was the bushings. Once replaced, everything was fine. Racing Beat did NOT supply the bushings for the MP3. Mazda did. The is warranty work folks. Bring it back.
The squeak noise in the back should only happen when it's cold. This is due to the type of suspension/shocks the mp3 has. Ask anyone that has added an aftermarket suspension to their civic or anything really, and they will probably confirm the same noises on theirs as well. Even a stock RSX does this as well.
The rattle in the door, if you have one can be fixed by taking the door panel off and putting it back on RIGHT! If it's not aligned properly, it rattles. Make sure it's tight too. I never had this rattle, but this is what someone told me.
It's not uncommon for ANY car to have rattles once it gets colder outside. We call them cold rattles here. They seem even more common on sportier cars. Once it warms up, they go away.
Personally, outside of the bushings being cold in the morning, the car is perfectly silent as it should be. Sometimes when it's REALLY COLD, there may be a cold rattle. Car warms up, rattle goes away.