I had that idle vibration when my car was new, it's gone now after 5000km. I think it's to do with the engine mounting being abit stiff while it's new. Once your car ages abit the vibration is no longer noticable, at least for me.

As for your tyres, 30 psi is way too low. The recommended psi for 16" is 36. Of course your steering feels heavy because you're under inflated.

The shaking at high speed is probably due to bad balancing and alignment. Mine wasn't perfect either, I brought mine to an alignment sifu and now the car is straight like a bullet and no shaking all the way to 200kph.

This is definitely not normal. I can't even feel any vibration in my car when idling. Car is also very smooth when driven at 100-140 km/h. When cornering or parking the steering should be light. I'm driving a Mazda 3 GL myself. The tyre pressure should be 36 - 40 psi depending on your passenger load (see label on driver door edge) but I don't think tyre pressure is your problem.  You definitely need to pursue this with the SC until solved.

Does Mazda service center provide free car wash when sending for 1st 1000km service?

The shop called it nano coating.. Products from Japan
Cost me around 2k+