Agree could be driveshaft or bushings, even something as simple as wheel balance and alignment, i would jack up and test ask those so called mechanics to place on ramp and test not guess and say they think its steering rack.
Induction kit will need to be custom made, unless you want to spend around £300, as far as I'm aware radio upgrade is android double din unit you may be able to modify your current unit to achieve this instead
not sure if saloon and tourer tailgate closer unit is same i believe not although check here to match up part numbers
https://www.parts-honda.uk/
It seem to appear tailgate closer is out of alignment
Some can be if you look around there may be caps, although my recommdation is to put on a battery conditioner such as ctek or optimate before replacing i have revived many batteries this way.
Yes i always use this where other chargers have failed does a trickle charge so can take few days for some batteries but 9 out of 10 have worked for me.