I think I found the trike you are talking about online. There is an awful lot of chain there to wear out as well.
A high-quality geared rear hub that supports a suitable tyre for comfort might be worth further consideration. Still some good choice in strong 26 inch rims because of downhill.
I...
Absent better testing gear maybe get a spectrograph application for a mobile phone and look for the resonant frequency in the audio spectrum. Better still clamp a contact microphone on somewhere sensible. It may not be some multiple of pole pairs and RPM. Maybe for example it is a harmonic of...
Patience, a wire brush and a small triangle file would be how I would attempt to clean up threads after removing mushrooming. There is a fair bit of clearance between a nut and a bolt and it just has to go on. Damaged axles for a vehicle will involve additional factors around safety.
If the wheel has spokes then the issue may just be what a wheel builder calls 'dish.'
If you take the bike to a bicycle mechanic they can check this. It is normal for a wheel to move a bit.
I once bought a pair of wheels for a non-electric bicycle. The front wheel was so far out I could not...