Pickups definitely make a difference. There are obvious big differences like going from single coil to humbuckers but also how hot or not the pickups are wired. Cheap guitars these days, well I mean Epiphone, Squire, that sort of thing, not dirt cheap, come with pretty good pickups now tough. So it's quite possible you won't hear that much of a difference but you just might too.
As for soldering, the other important thing aside from all the good points mentioned above, is to make sure there is wire to lug contact. You want to solder the wire to the metal lugs making sure you have contact and not have the wire just floating in the center of the lug held together with a blob of solder.
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