Should there be "minimum" and "maximum" sentencing requirements in law? Or should courts have discretion? Which is better for a "justice system"?

Mandatory minimums are a problem. Judges lose discretion to tailor the punishment to the specifics of the case. Minimums may be pushed unreasonably high so politicians can claim to be "tough on crime." (This happened big time in the US, starting with the War on Drugs in the 1970s and continuing through the 1990s.) Both of those lead to more people in prison longer than they should be.

Also, at least in the US, not all crimes carry mandatory minimum sentences. This gives prosecutors a new source of leverage:

The use of mandatory minimums effectively vests prosecutors with powerful sentencing discretion. The prosecutor controls the decision to charge a person with a mandatory-eligible crime and, in some states, the decision to apply the mandatory minimum to an eligible charge. Rather than eliminate discretion in sentencing, mandatory minimums therefore moved this power from judges to prosecutors. The threat of mandatory minimums also encourages defendants to plead to a different crime to avoid a stiff, mandatory sentence.

https://www.sentencingproject.org/fact-sheet/how-mandatory-minimums-perpetuate-mass-incarceration-and-what-to-do-about-it/

Mandatory minimums can also lead to significant racial disparities. The linked article cites an example of very different minimum sentences for different drug offenses, leading to a sharp rise in incarceration rates for blacks but much less so for whites.

The use of mandatory minimums effectively vests prosecutors with powerful sentencing discretion. The prosecutor controls the decision to charge a person with a mandatory-eligible crime
Wrong problem. Minimums and maximums are a bad workaround for this.
Obviously the prosecutor should not be the only person who can decide which crime to charge, or not to charge, and this decision should not be possible only at the beginning of the proceedings.
In our country the judge can easilly change the exact crime(s), and this is done very often, because it makes everybody's life easier and the truth is served much better when this decision is made again after all evidence, witnesses etc. have been seen and heard.
- Good point. I will add that to the long list of reforms we need in the US criminal system.

Justice shouldn't he punitive, it should be rehabilitative. For that purpose, jail should only be used when there is no chance of rehabilitation and the person needs to be separated from society because they're a danger to society.

I think there's absolutely a middle ground where someone is currently a danger to society, but they're capable of rehabilitation. However, we absolutely need to continue that work on the outside once it's safe for them to be released if we want to reduce recidivism of violent crime.
I think it should be both. Punishment can be a good deterrent, and it's really not possible to measure how much of an effect the deterrence has. You don't have statistics on people who would have committed a crime if there weren't harsh punishments.
I know it's off topic but what has been proven to work best is to prevent crime by improving the lives of would-be criminals before they get to that situation. It's cheaper and just better in every way.
- Regarding the first paragraph, the way they measure this is observing the incidence in different circumstances. Similar country but higher punishment? See if fewer people do the crime. Oversimplified.
  The research shows that the deterrence effect exists but, beyond a certain punishment level, it doesn't do much anymore. What helps is primarily the odds of being caught at all (and then an appropriate punishment) and secondarily the time between violation and punishment (I didn't realise this would matter for adults but apparently so)
  Going to jail for two years, four years, or six years, either way you lose your social life, the roof over your head (once you get out), your job, everything. It's a doubling or tripling of the sentence but is it really that different? If I'm okay incurring 2y prison sentence... I'm probably not the target audience for this but I imagine such a person would also risk 6y if they want someone gone that badly and the odds of being caught are low enough
  Prevention is golden, as you say. But then rehabilitation is silver imo: if they lose everything, feel thrown out by society, what still drives them to do good afterwards? I'm sure many of them will simply want to better their lives but external motivation must also help
  So I see it like the people who I saw saying upthread that rehabilitation should be the goal: if they're a danger to society, idk, whatcha gonna do but control that? Need to lock them up or similar (ankle thingy, idk). But if there's a good chance they'll get back on their feet and become taxpayers instead of prisoners or crime group members, then that's what we should asap strive for

Sentencing to long prison sentences should be avoided. There should be two general categories:

danger to others
not danger to others

In the danger case the sentenced should be kept imprisoned until proven safe, otherwise only a light deterrent is needed.

There isn't much difference in the deterrence between 1 year and 10 years for most people. The really violent crimes are committed either without regard for oneself, or as a part of organization that already has a support system inside the prison.

Crime motive is subjective, judges are subjective, justice is an ideal that is served by mins and maxes. I can’t fall on either side of the question totally but I do think objective punishments for objectively proven crimes seems more just.

I think without them, the disparity in sentencing between race and gender would be even worse than it is (and it's BAD), so... Probably a good thing overall?

Minimum sentences sounds like pretty bad. There is cases of legally wrong, morally right. Think about killing your violent spouse which while not being within the limit of self-defence is pretty different from killing your spouse. It's the job of criminal lawyer and prosecutors to come with reason to give a lighter/harder sentence and the judge job to listen to this arguments, compared with similar cases and sentence guideline to give the fairest sentence possible.

A person who seek therapy, pay penalties to the victim, has now a job, and already try to be a better person at the time of sentencing has nothing to do in jail

What about unjustly light sentences though? Like a sex criminal who is let off easy because he's white, has rich parents and was on track to get a high paying career?
- Then the prosecution brings case law where a person of color in a similar situation got a very different sentence, or someone without the rich parents etc.
  I'm sure there'll be biases but minimum sentences won't undo those. I'd find minimum and maximum sentences very unfair for situations like described by the person you replied to. Case law and perhaps a blinding system (Justice is blindfolded after all) where at least one of the judges involved doesn't get to learn things like "white guy with rich parents" might be a better solution if that's the problem this is intended to solve
- Or someone can just skip prison and become leader of the country instead ahem USA ahem

There needs to be some parameters just to prevent someone from getting a slap on the wrist for a major crime or sent to the gulags for a minor offense.

Why would a judge do that in the first place?
Or if they do, what prevents either side from going to the next court to get it overturned? And the next court after that
Idk if this is everywhere but the system I'm used to is this: the first two levels of court look at the case details in-depth, then there's a third 'last resort' court for if you think there was a mistrial (this usually gets rejected), and they can send it back to the second court to re-do if something crazy happened that's not in line with correct procedure as it sounds like it did in your example
If we don't trust a whole series of judges to pass judgment fairly, then I'm not sure we should have judges. Personally I trust these more to apply laws and case law than if we'd put elected politicians on the seat of judge, as basically happens with them choosing the sentence parameters (and as you see more and more often in my country; older judges also speak of higher and higher sentences being expected, makes me wonder if we'll go full-circle to medieval practices eventually)

Minimums help the judges not only to decide the punishment, also to see in advance which crime is worse than the other crime. Maximums could maybe help in a similar way.

Maximums are needed when stark punishments exist, like torture, death penalty or removing of body parts. These can be limited to specific crimes then.

I would expect that judges have a fine moral compass for which crimes are worse than others. Better if they don't need to follow some administration's counterscientific "tough on crime" whim
You make a compelling case for maximums though. I hadn't thought of it as being a way to reduce torture or death penalty in the countries where that's on the table in the first place

Court discretion, many factors may influence a sentence and if you don't trust the judge on that, I would say you have a much bigger problem in your hands.