I've got a question about cheap guitars where the knobs have almost no effect (or there's a 0 and 10 setting, no in between) and the strings go out of tune often. Can these problems be fixed by resoldering the pots or screwing in the tuners tighter or something along those lines? Or do these parts have to be replaced to be satisfactory?
Got Bass?
replacement, cuz i have one like that (it's my beater), and it has a similar problem(s). I've tried stuff with the tuners, and they're just really cheap, its not how they're put on, and the pots aren't very good.
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replacing the nut on a cheap guitar can help it stay in tune much better.
Cheap parts are just cheap and low quality. A nut helps with tuning sometimes if thats what it is. But on cheaper guitars they use the noname asian tuners which are pretty much junk. And of course will use the cheapest electronics they can find. Replacement is usually the only way to fix such things.
Usually the part itself needs to go. Really cheap tuners, electronics, etc are often incredibly mediocre. If you are using a tremolo or doing a lot of heavy bending, having a good nut, or at least making attempts at using graphite to lubricate the slots can help your tuning stability.

As far as pots go, cheap ones are just cheap. Get a new set of at least higher end alphas or maybe some CTS ones (logarithmic taper for volume, linear taper for tone)
I replaced the nut with a bone one on my epiphone SG special and it now stays in tune much nicer
Will lubricating the nut with graphite really help it stay in tune? Wouldn't lubrication make them go out of tune more?
Got Bass?
If you are using a tremolo then you want the string to properly return to its original position after being slackened or stretched. Maybe using a bit of pencil lead (graphite) will help, or I know they make nut lubricant, or you can just go and get something like a graphtech nut.