Why did Malvo do the things he did to Stavros Milos?

I saw that Malvo gets consistently portrayed as entirely vague and entirely evil. The only solid characterizations for him are the cassette tapes he carries around of all these recorded conversations. It's hinted at a lot, but when we see him living as the dentist, we finally see an unequivocal scene where he is just sitting alone, listening to the tape, and having some kind of existential, sadistic joy at the horrific circumstances that Lester caused by killing his wife.

There was always a thought that perhaps the tapes are insurance, for blackmail, but I think this scene after his dinner party (as the dentist) shows that he really is just keeping them and listening to them for his own pleasure.

So if you keep this in mind and look back on the circumstances surrounding Milos, he saw an opportunity and he took it. I think it's nothing more than his own pleasure, to sadistically mess with this petty, dumb, and selfish man.

Originally he's there on behest of the crime organization he works for, and he's supposed to determine who the original blackmailer is. But when he realizes how dumb the whole situation is, and how he's been brought in to solve this "problem", it simply becomes a game for him.

I think it'd be too forgiving to ignore that this also ate up a LOT of the season's episodes, and it was a convenient way to write in these crazy situations while Lester's character was fleshed out and was digging deeper and deeper into his own sadistic insanity. Also it kept Malvo around Bemidji and Fargo, so that he could run into Wrench and Numbers, and Molly, and continually mess with the entire situation.

Because the whole thing started when he hit a deer and had to kill a cop and get to the hospital, where he meets Lester. He doesn't know or care who Sam Hess is, and doesn't even realize that his own organization is sent out to kill him for killing Sam Hess. Malvo's just sorta doing whatever he wants.

So, I think from the thematic sense of the show, the heart of the plot is really the first two or three episodes and the last two, and everything happening in-between, while I admit I loved it, is thematically sort of stretched-out for "fun". We get all the crazy characters and all the random circumstances and the build-up of coincidences, so that everything can finally come together for the final episode.

Milos was essentially just a big distraction, and if he could get a huge payday out of it, then it's doubly-enjoyable. And note that the show takes place in 2006 and Malvo goes around with essentially the same clothes and the same walkman and cassettes. He certainly didn't care about money or things, so he was doing it all to entertain himself in his sadistic nature.

Malvo is an extremely odd character. You can say he is somewhat similar to the Joker in The Dark Knight. These are psychotic criminals without clear motives for a crime (Joker, obviously, being way ahead of the curve). There are several instances from the show where Malvo does strange things to people. Some examples :

1. when he check in to a motel, he provokes the motel bell boy to piss on owner's car. But when he is actually doing it, Malvo calls the owner which gets him caught.

2. In last episode, when malvo is searching for Lester's address and meets the new occupants of Lester's old house. He scares the kids by telling them false stories.

Also think of it this way, why did he kill Sam Hess in the first place. Malvo had nothing to do with Lester or with Sam Hess.

Malvo is a psycho criminal in its true sense. In order to understand psycho we should take reference from the Joker. In The Dark Knight, Batman also becomes confused with Joker's intentions and motives and then Alfred tells him a story about a bandit in Burma who used to steal rubies but throw them off afterwards. Alfred sums it up as:

Some people just want to see the world burn

I think, this piece from The Dark Knight can be used to understand Malvo.

So to answer your question, "why did Malvo do the things he did to Stavros Milos: because he is a psycho. He often likes to hurt people without any reason.

Spoiler Alert

He doesn't kill only bad people, as the elevator scene (and others) show. He saw Milos as prey to be toyed with. Malvo is certainly a psychopath, he seems to enjoy manipulating people, seeing them as only a means to an end. For example, in season 1, episode 10, he laments to Lester, of the work he put in to master his new identity as a successful dentist, “I worked this guy for six months, Lester, six months. Can you imagine the number of sewer mouths I've put my hands in, the gallons of human spit?"

He considers himself a predator, often appealing to his animal nature, such as when he says, "there are no saints in the animal kingdom, only breakfast and dinner" (Season 1, episode 6). In particular, I think he sees himself as a wolf. There are numerous on-screen references to wolves, including the wolf head that is mounted on the wall of his cabin, and the wolf that crosses paths with Colin Hanks' character, foreshadowing the show-down between Malvo and himself (Season 1, episode 10). In season 1, episode 6, Malvo says to Milos, "the greatest empire in human history [Rome, was], founded by wolves. You know what wolves do. They hunt. They kill. It's why I never bought into the jungle book. Boy is raised by wolves and becomes friends with a bear and panther. I don't think so."

Of course, Malvo's reasoning is a flawed post-hoc justification for his actions. Wolves hunt and kill in order to eat, Malvo does not. As others have suggested, Malvo is pure evil, not unlike Vautrin in Balzac's novels (who, like Malvo, impersonates a priest in order to manipulate his victims).