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 Lew found him sympathetic . . . the oblique planes of his face revealing an origin somewhere in the Slavic vastnesses of Europe as yet but lightly traveled by the recreational visitor. . . .

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"Lew found him sympathetic."
An ambiguous statement. Was Lew sympathetic toward Max or did he note that Max expressed sympathy toward others? My guess is that Lew felt drawn to Max, probably due to Max being, in a way, banished from his life in Europe, just as Lew had been banished from his past. Both were jettisoned from the previous arc of their life to crash land here in Chicago and had found themselves working security detail on the Archduke Ferdinand.

"Slavic vastnesses of Europe as yet but lightly traveled by the recreational visitor"
Max was born in either a rural or dangerous place, perhaps a bit of both. Anyway, somewhere off the beaten path, some place tourists either wouldn't know about or wouldn't dare wander for safety concerns. As for the "Slavic vastnesses of Europe," I suppose that just means he was born in one of the various Slavic populations spread across Europe. You'd expect the Austro-Hungarian Empire, being that's where he's now employed, but who knows, right?
    Perhaps this mystery of Max's origin, at least to Lew's estimation, also causes Lew sympathy toward Max, being that Lew also comes from a mysterious past, albeit less geographic and more amnesiac.

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 Sending him to America seemed appropriate.

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I don't know if I should feel insulted by this paragraph concluding sentence or I should break out in a patriotic cheer! "This guy is too nuts to keep around but he's also too nuts to get rid of. Where could we send him where he wouldn't stand out?" Um, go America, I guess? It's like if Waldo from Where's Waldo? was just a drooling maniac in a torn up straight jacket with a smoking gun in one hand and a bloody cleaver in the other. You'd need a picture with similar looking people to hide him well. And, apparently, that scene is Chicago, U.S.A.

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 Despite his youth he was said to give an impression of access to resources beyond his own, of being comfortable in the shadows and absolutely unprincipled, with an abiding contempt for any distinction between life and death.

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Max Khäutsch is the Darth Maul of Against the Day. That means he's both a tremendous bad-ass and, I'm predicting, he's defeated in a humiliatingly brief battle.

I'm just talking about The Phantom Menace Darth Maul and not all of Darth Maul's subsequent appearances and build up of his character simply because some marketer was all, "Wait. The most loved character in this movie died immediately? What are we going to do about that." And somebody shrugged and was all, "Not much we can do! Kenobi cut him in half!" And then a mahogany table full of suits sat around brainstorming, mostly going, "Um. Uh. Duh?" Then one of them was all, "Wasn't Darth Vader, like, 85% robot?" And somebody snapped their fingers and was all, "Yes! Darth Maul can be half robot! Now, which half should we make robot?" After that they worked out the details and stuck him in some cartoons.

Just in case my comparison with Darth Maul doesn't mean anything to you, let me say that Max seems like a savant when it comes to criminal activity and assassination. Everybody says, "He's so young! But his contacts with the underground and his knowledge of the 1893 dark web are astounding! Unparalleled! This boy can do anything he wants, especially since he's unconcerned with how many people are standing between his target and the wildly erratic sword he's swinging!"

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 Standard Hapsburg procedure would have been to put him out of the way at some agreed-upon point of diminishing usefulness, but nobody was willing to try.

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If I'm following, Max Käutsch was seen as too dangerous to keep around, based on his deadly assassin abilities, which would normally mean he'd be sentenced to boring bureaucratic jobs far from urban centers of any importance. Perhaps that's the kind of thing you do with a government employee who was used to murder rivals or dissidents, an employee you just can't have in the public eye, or being seen associated now with the heads of government or members of royalty. It would be like if the photographers who chased Diana's car down, causing the crash, had been working for the royal family. They'd immediately want to distance themselves from them, for propriety's sake. Also for the sake of they obviously hired them to hound Princess Diana to death but they didn't want it to seem that way, so best send them to the Isle of Wight to take family portraits of the locals instead.
    More probably, I'm misunderstanding this sentence. Pynchon is sometimes difficult, especially for somebody who has trouble with the concepts in Clifford books. Why is that dog so big? How does the family afford feeding it? Do they need a pick-up truck for his doody when they go for walks? How hard is it to ignore Clifford's lipstick when he's giving himself a good lick?

If I've got the gist of this right then that means I understand the usual procedure for dealing with somebody like Max in your ranks. But since Max is such a scary bad-ass, nobody has had the nerve to push him out of close proximity to the Hapsburg family. And that's why he's become the captain of the unit in charge with keeping Archduke Ferdinand safe during their overseas travel. Sure, this job gets him "out of the way" of the Austro-Hungarian Empire. But he's still in close proximity to Ferdinand, and he still retains major responsibilities.

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 Young Max Khäutsch, newly commissioned a captain in the Trabants, was here on his first overseas assignment, as field chief of "K&K Special Security," having already proven himself useful at home as an assassin, an especially deadly one, it seemed.

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Now I'm really confused by what Lew meant by his "opposite"! Maybe he just meant the guy leading the protection detail on the Austrian side, just as he was leading the protection detail on the American side. That take seems to be justified by the evidence here, where Max Khäutsch is the captain of the Trabants. The evidence against Lew looking for somebody like-minded is that Max is a proven assassin who has apparently killed multiple targets. What has Lew done? Divorced his wife and gone through Interdimensional AA? I suppose Lew has done something else super terrible but it's not like Pynchon is ever going to reveal what that was!
    Max being "an especially deadly" assassin probably means he's as excited about hunting Hungarian Stockyard workers as the Archduke and Nate Previtt. Come to think of it, I can't think of a single person in 1893 America who would have been against it aside from the Hungarian Stockyard workers and who was going to listen to them?! Maybe suffragettes? But then they had their own problems!

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 Curious himself about who might be his opposite number on the Austrian side of this exercise, Lew nosed around and picked up an item or two.

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What does Lew mean by his "opposite number"? Does he mean some Trabant on the Archduke's team who thinks like he does? Who thinks this whole "hunting humans" idea is not just grotesque but so absurd that it shouldn't even be regarded? That's my guess. Lew's looking for somebody who isn't completely insane and/or racist to back up him up if the "hunt" should even begin to move forward as a possible thing.

But the word "opposite" could also mean one who has completely contrary thoughts to Lew's. Which could mean he wants to find the guy most gung-ho for this idea so he can beat him to within an inch of his life and leave him in a garbage can in an alley. I'm not sure Lew is capable of that but he's become a noir detective and I'm fairly certain those types are capable of that.

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 "With more them damned anarchistic foreign-born south of Forty-seventh than you could point a Mannlicher at," chuckled Nate, "sure'd be a few less of em to worry about, wouldn't it?"

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This is the kind of thing every conservative American who stridently claims they're not racist when accused of being racist would eagerly say in one of their "safe spaces" (a space with a bunch of other white faces, all of whom they automatically believe are as "not racist" as they are). This quote by Nate could easily have been lifted off of right-wing Twitter just this morning, with a few minor substitutions.