Why did Moses have the sons of Levi run about with swords slaying family and friends?

I would say the main thing to remember here when dealing with the Old Testament is that the wages for sin, no matter the sin according to God, was death. So I believe you answered your own question as to Why they were told to slay their brethren. That is was indeed to atone for the sins of the people. Now according to the passage I posted below it says that Moses declared that whomever was on the side of God was to come foward. So for all intents and purposes I'd say they volunteered. Perhaps they were the only ones grounded in their faith enough to shun the Golden Calf and refused to worship it. For whatever reason God chose them to bestow His wrath on the nation of Israel. The text doesn't really say why it was the sons of Levi.

"Now when Moses saw that the people were unrestrained (for Aaron had not restrained them, to their shame among their enemies), 26 then Moses stood in the entrance of the camp, and said, “Whoever is on the Lord’s side—come to me!” And all the sons of Levi gathered themselves together to him. 27 And he said to them, “Thus says the Lord God of Israel: ‘Let every man put his sword on his side, and go in and out from entrance to entrance throughout the camp, and let every man kill his brother, every man his companion, and every man his neighbor.’” 28 So the sons of Levi did according to the word of Moses. And about three thousand men of the people fell that day. 29 Then Moses said, “Consecrate yourselves today to the Lord, that He may bestow on you a blessing this day, for every man has opposed his son and his brother.”-Exodus 32 25-29

On an interesting side note, the end of the passage has a vague reflection of what Christ would do and does in our society today. Christ or rather the belief in Christ divides households, tears relationships apart, and leads to persecution. Believers will always have the minority and thus be outcasted until the Lord's return.

Hope this helped, God Bless

The answer to the first part of your question, appears to lie in the fact that only the Levites responded to Moses call for all who were on the Lord's side to come to him.

It is also good to understand that not all who rebelled against God on that day were killed, which may answer your, second question about the added punishment which was meted out to all the offenders. The estimates of the number of Israelites leaving Egypt has been numbered in the Millions, but only about three thousand were killed.

All Scripture is quoted from the New Revised King James Version.

Exodus 32:25 through 29 And when Moses saw that the people were naked; (for Aaron had made them naked unto their shame among their enemies:) Then Moses stood in the gate of the camp, and said, Who is on יהוה's side? let him come unto me. And all the sons of Levi gathered themselves together unto him. And he said unto them, Thus saith יהוה Elohim of Israel, Put every man his sword by his side, and go in and out from gate to gate throughout the camp, and slay every man his brother, and every man his companion, and every man his neighbour. And the children of Levi did according to the word of Moses: and there fell of the people that day about three thousand men. For Moses had said, Consecrate yourselves to day to יהוה, even every man upon his son, and upon his brother; that he may bestow upon you a blessing this day.

We also have to consider that the Levites had been taken by God in place of the first born of Israel when he killed all the first born of Egypt, but spared the first born of Israel. The Levites were specifically chosen to be the servants of God, and were set aside from the other tribes to that purpose alone.

God could just as easily sent angels to vent his ire upon the Israelites, however, considering that the Levites were also dedicated to God's service as are the angels, what God may done was to test the Levites to see if they were as dedicated to him as are the Angels. That is not Scripturally based is my own observance.

We are not told whether or not the Levites took part in the revelry and worship of the 'Golden Calf ' or not, but logic tells us that not all Israelites took part in the rebellion, however the fact that both Aaron and Merriam survived the slaughter says that God did not punish everyone taking part since Aaron was the one who formed the Calf to start with.

We also need to remember this;

Rom 9:15 and 16 For he saith to Moses, I will have mercy on whom I will have mercy, and I will have compassion on whom I will have compassion. So then it is not of him that willeth, nor of him that runneth, but of יהוה that sheweth mercy.

Finally do we deserve God's mercy any more than those who rebelled against God?