A bit of wobble is tolerable on cheaper hubs, but that is a excessive. Probably very difficult to adjust the shifting, if able at all.
Depends on where the wobble originates from. That much and the way it behaves is probably misadjusted spokes and not the axle or bearing. I'm 70% sure you could fix this by adjusting the spokes. There's probably a tutorial somewhere. If the rim is too crooked, you might have to straighten that out first, but it usually isn't necessary.
You will need a bit of patience if you've never trued up a bike rim before. No idea what a repair shop would charge for doing it for you, but check what a new wheel would set you back first.
100% get the wheel trued @ a shop if you have never done this procedure before.
Thanks. Just did a check with a zip tie; the rim oscillates all over the place, it's not like there's just a small region that's off and needs adjusting. I don't think fixing it is feasible, so I guess a new wheel it is.
Play in the hub bearings is not great because it leads to uneven loading of the bearings and likely faster degradation.
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