US-based Bose is certainly working on a system that will cancel out noise in cars. At the moment, though, it’s not available and will probably only be fitted to brand-new cars as standard equipment. At first anyway, because tech like this has a habit of tricking down to the aftermarket. However, that’s not much help to you right now.

Wearing noise-cancelling headphones is not practical (or sensible, or legal) so you need to look at an old-school solution such as the tyres you mentioned or stripping the car’s interior and fitting a noise-suppressing matting under the carpet and inside the doors. If it’s any consolation, it’s not your imagination; Mazdas for the last 20 years have been harshly criticised for their interior noise levels when the competition seems to do it better.

Ironically, active noise cancellation technology as proposed by Bose is not to tame mechanically noisy cars, but to remove wind and road noise from electric cars which are otherwise so quiet, these secondary noises become a big nuisance.

It’s likely to be one problem that is causing all of the issues you mention. It needs a diagnostic check to locate the cause.

You don’t say what makes it hard to get in and out for you, so I’m taking a stab in the dark. There are a couple of possibilities, one being that the doors are too short and don't open wide enough, another is that it’s too low. A move to a mid-size hatch like a Mazda6, Honda Accord or Subaru Liberty would probably address the first, while a high-riding SUV such as a Kia Sportage, Hyundai iX35 or Mitsubishi ASX would help with the second.