As some of the others have stated, this is overpowered like mad. Not a bad first try, but you need to reign this in big time. Gonna spoiler individual issues to make this post more readable. Note that I've only lightly scanned other posts.
Proficiencies
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First of all, this class starts with free proficiency with the spiked chain. Alarm bells going off now: stick to simple/martial weapons. If you want, limit the martial weapons to bows and finesse-able weapons, though I see no real need to do this. Unless the class has some exotic weapons that are specifically thematic to it (as Monk, Ninja, Bard, and the like) I see no reason to make them free.
Death Step
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Death Step in itself isn't a bad idea. A good concept that could very well carry the class. As you've implemented it, it's just insane. First of all, it absolutely shouldn't be unlimited at any level. Whereas the previous posters have suggested making it take actions and get more action efficient as you level, I suggest leaving it a flat swift action and give it a limited number of uses per encounter. Start at 1/encounter at level one, and then get one more every level divisible by 5 (5, 10, 15, 20), off the top of my head.
Most of the other abilities that come with death step need to just go away. Even adjusted as I mentioned above, the ability is already great and enough to carry the class: making you invisible and blurred and displaced and all that as well is nine different kinds of overkill. At the very most, give it a blurring effect at high level (13+). If you've limited the number of times per encounter the Slayer can use the ability, maybe make it last multiple rounds. (3 sounds good to me.) Beyond that, we quickly start getting into overpowered territory.
Swift Strike
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Swift Strike as-is is just massive packets of damage. I pick Diamond Mind for my third discipline and my favored discipline. Level twenty, initiate Raging Mongoose boost, then hit with Time Stands Still. Let's be conservative and say I'm using a basic 1d8 rapier. (1d8+7d6) 12 times, average 29 damage per strike. Before any other modifiers at all, you do ~350 damage on average with that. Presuming your rapier negates his DR (not difficult at that point), congratulations, you just one shot a freaking Balor, and you're not even trying hard. You even have plenty of damage left to compensate for a few misses.
My first thought upon reading this was, "wait, isn't this just the Scout's skirmish?" I suggest looking up Scout, and replacing Swift Strike with a (much) slower progressing Skirmish. Off the top of my head, progress half the speed of a Scout's skirmish, ending with an entirely reasonable +3d6 damage, +2 AC at level 18 or 19. This also has the added benefit of letting the class benefit from support for the Skirmish feature included in other books (ambush feats, prestige classes, and the like).
Combat Styles
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You could do worse things here, but the progression is way fast, in that it's identical to the Ranger. With all the other things this class has going for it, it's kind of silly to completely ape the Ranger's shtick with this. Maybe reduce the progression so that it comes later: the first level doesn't come until level 3, the Improved until level 9-11, and Mastery until 17-20.
Other than that, the styles aren't particularly offensive on their own. For Two-Weapon Fighting, I suggest reducing it to a single extra attack in conjunction with strike: otherwise you get full off-hand attacks from doing a standard action strike, which is silly and overpowered. For Combat Style Mastery (Thrown), I suggest increasing either the range OR the multiplier, not both.
Quick Perception
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What the hell is this even supposed to represent? It comes out of nowhere, I can't think of any reason at all for this class to have that ability. You've got enough going for you already. If you really want something like this, take the Diamond Mind stance Hearing the Air.
Called Shots
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Think about this for just a few moments and I think you'll see how this is a terrible, terrible idea. The most offensive here is headshots--for a -6 attack penalty (not a big deal on your first attack: I see heavies power attacking for this much regularly at these levels), you can completely stun lock anything not immune to stuns indefinitely. This should scream overpowered: they don't even get a save.
The other targets are quite abusable too. A called shot to the arm can reduce any non-caster to a helpless lump for four rounds straight. Leg shots are a perfect shut-down in whatever form you like that are crazy control in a class that doesn't really need it. Chest strike can be hella abusable if, say, I'm power attacking in addition to the called strike (granted, I need some help from a buffer to make that work due to the attack penalties, but presumably you have allies helping you here).
My suggestion is to just end this ability alltogether. This class has more going for it than any other initiator class already. [edit]Adding saves takes them from HORRIDLY overpowered to just regular overpowered. I still think they need to go.
Misc. Other Things
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If Weapon Finesse remains a feat tax for the Rogue, Ninja, Swordsage, and basically every finesse fighter but Swashbuckler, it should remain one for this guy. I would take that out completely.
This class deserves Evasion, but I'm not convinced it should come as early as you've got it right now. Evasion could fill in a dead level somewhere in the 7-9 range. Swordsage doesn't get it until level 9, so use that as a comparison.
I don't have a problem with the wall running part of Acrobatics, but that skill boost is massive and unnecessary--if the Slayer wants to be acrobatic, let him just put points into those skills.
As with Evasion, Improved Evasion comes too early. Fill a dead level somewhere in 16-20 with it, I think.
Freedom of Speed needs to just be scrapped. Get a mage friend or buy the ring.
Unknown Movement can just be replaced with the Darkstalker feat (Lords of Madness, forces creatures with those abilities to make a Listen or Spot check versus the user's Hide check), which is slightly less potent but does the same thing. It seems like it would be a big deal intuitively, but I've found in practice that it doesn't actually come up very often, so I see no real need to change that.
Masterful Finesse should go. If Swordsages need to take Shadow Blade to get dex to damage, so should Slayers.
Quick Counter needs to go. There are actual counter maneuvers that do this just use them.
Death Strike needs to go. Everyone else needs to take PrCs to get this ability. If a Slayer wants to do death strikes, he can easily qualify for Assassin.
Superior Precision is out of place and the power potential is insane. Recall my Raging Mongoose Time Stands Still strike example above? This could make it perfectly accurate as well. Axe this.
Perfect Speed needs to go. There are stances that replicate this: use this instead.
Deadly Speed needs to go. First I do a Raging Mongoose Time Stands Still strike, and then I do a Girallon Windmill Flesh Rip full attack. I go from one shotting a Balor to one shotting the Tarrasque.
I'm being harsh because I see potential here, but you fall into a common trap of wanting this class to be so awesome that you lose any sense of balance. It happens to the best of us, but the sooner you curb this, the better a homebrewer you will be.
[edit]It looks like I posted in the middle of you editing. Reading the edit now.
[editedit]Rogues can make as many sneak attacks per round as they have attacks. Same for Skirmish, if you can find a way to move and make multiple attacks. You can go the 'one attack per round' route, but I think just going with Skirmish makes more sense.
Most of the comments I made before still apply, so I'm not making major revisions to my post. Just an occasional touch-up.
Re: The Swift Slayer [3.5 base class] PEACH appreciated
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