Aside from lowering tyre pressures(not a great idea), any increase in size of rim from the factory 16 inch will make the ride harsher. The AT-X is not a rough car from factory, so I am unsure of what ride quality you are expecting.
Whilst that method is fine for engine mounts, it should definitely not be used for suspension/steering components.
If you are unable to source bushes from Whiteline, Nolathane or Superpro, then genuine will be your only legal and safe repair.
They both could.
Just take it to Pedders or similar and have a safety check done, they aren't very expensive. Let a professional diagnose it in person for you.
If you then feel the need to DIY, attempt to replace them yourself.
The "rear strut" is the rear axle. The sway bar is bolted to the underside of this.
The longitudinal rods you have drawn are most likely the trailing arms.
The latitudinal rod is the panhard rod(although it may have other names). It appears that the bushes could be new(Whiteline) from your photo.
Poor maintenance will definitely shorten its life, regardless of the engine type. Given the previous issues it is not a worthwhile risk, especially if it is at the stage of affecting performance.
They are a "sealed for life" transmission, which is a very loose term. Get it to a transmission specialist ASAP, not your dealer. A flush and oil change would be the cheapest place to start.