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They are charged with both because they killed the person somehow.  That way, since they are definitely guilty of one of the charges, they can't get off by claiming that they were charged with the wrong one.  It would be pretty dumb if someone charged with manslaughter could get away with it by saying "I killed the person deliberately so you have to let him go".

Easy.  There's nothing to say the prosecutor can't bring different charges that are substantially the same, and as you said, these ARE different offences.  This is done because it gives the jury the choice.  They might decide the killing was not deliberate so they're going to say not guilty of murder, but he did cause the death so it's manslaughter.  If the rules of court are that you can't be convicted of something you weren't charged with, then the prosecutor isn't going to bring just a murder charge if it could be either.  Otherwise you could have the jury letting him off completely if they think it's manslaughter but he wasn't charged with it.

You would, of course, expect that the jury will find the accused guilty of only one of the two, or neither.  And the judge will explain to them what their choices are.  That's actually his biggest job in a jury trial apart from playing referee - being the jury's legal adviser so they know what to do.  "If you think this, then it must be murder, but if you think that, then you must convict of manslaughter."  Been there, done it - the judge's summing-up speech at the end is really important.  "You have heard the evidence, now this is how to think about it and what choices of verdict you can make."

In English law, the Crown prosecutor can bring a murder charge and the judge can instruct the jury that he will accept a verdict of murder, manslaughter or not guilty.  So there is no need to bring both charges.  Similarly for assault - there can be GBH with intent, GBH without intent, ABH or common assault, and the Crown prosecutor will simply bring a charge of the most serious one it could be, knowing that the jury has the choice of making it something less and the judge will tell them what their choices are.

But if you don't have rules like this, the prosecutor must bring charges of all the things it could possibly be or the killer could be let off on a technicality.

Just out of interest, I did jury service on a trial where there were 17 charges of rape and indecent assault at different times on 4 girls.  There was a real question of whether what he was accused of was actually rape.  How much sexual penetration makes it rape? - what the girls were saying was borderline, and it had to be explained to us exactly where the border lies.  We could have decided there were no rapes, "just" a number of assaults that were nearly rape but not quite.  If there were only rape charges, he could have been not guilty of anything.  Of course if we found him guilty of all the rapes, we had to find him guilty of all the assaults as well (and we did), but it had to be done that way to "cover all the bases".

The Minn laws are similar but not the same, Murder 3 requires

609.195 MURDER IN THE THIRD DEGREE.(a) Whoever, without intent to effect the death of any person, causes the death of another by perpetrating an act eminently dangerous to others and evincing a depraved mind, without regard for human life, is guilty of murder in the third degree and may be sentenced to imprisonment for not more than 25 years.

While manslaughter is:

609.205 MANSLAUGHTER IN THE SECOND DEGREE.A person who causes the death of another by any of the following means is guilty of manslaughter in the second degree and may be sentenced to imprisonment for not more than ten years or to payment of a fine of not more than $20,000, or both:(1) by the person's culpable negligence whereby the person creates an unreasonable risk, and consciously takes chances of causing death or great bodily harm to another; or ..........

They charge with both so they get a conviction on at least one.  If you try for one or the other, you cannot retry, double jeopardy.

Murder three is a different charge. Last person I know of to be charged in my area(my State being one of the places that still has it) was charged because he was just looking for someone to kill. He was driving around a neighborhood. Someone havinga party yelled at him to slow down. He stopped, grabbed his gun and went and killed the guy. Didn't know the guy or anything, just had a gun and was looking for someone to kill. Got Murder 3 because it was more severe then manslaughter, but not severe enough to kill anything else. So based on the timing I'm guessing you are thinking of that cop. Well I don't think murder 3 is accurate. TBH I think if this GOES to trial and the charge sticks he will ask for a bench trial. A Judge would see that this isn't murder 3 and would convict on manslaughter in my opinion. However people are angry. The DA has to try in this case. Saying we're going to charge him with murder, even on a weak case, will help. In other cases as was said Murder 3 and Manslaughter will both be charged because of how similar they are in legal definition. The way I was taught was voluntary manslaughter was when you didn't intend to kill them but you were aware your actions COULD kill them. Like you punch a guy and he falls back and hits his head on the cement and it cracks his skull open. Your intent was to do harm, just not that serious of harm. The line can get blurry though between 3rd degree murder and manslaughter with just how much harm did you intend to do.

To give the jury a lesser charge to go for if they don't think the facts of the case meet the threshold for murder. 

yes, in some states this is normal.  then the jury decides fi the situation rises to murder or not

If a single act COULD meet the legal definition of more than one crime, it is actually NORMAL to charge the person with the most serious, and at least one of the less serious crimes.

In the case you suggest, if they jury was unable to unanimously find that the intent required for murder had been proven, they could still convict for manslaughter.

It would be rare to actually convict on both charges for a single death.

Prosecutors charge both to give the jury the option of convicting on a lesser charge if they don't believe it can be proven they are guilty of the higher charge.  In Minneapolis, Murder 3 and manslaughter are basically the same charge, many states don't even have a murder 3 on the books thus automatically go with manslaughter