Most of your suggestions are good, but others seem to be forgetting something... Since I have time now, I'll address everything so far. First, some more information: I may not be the one using this character. I've started playing this past 4 weeks but I love the character creation so much that I've just been making more and more characters. Due to this, I've opened my "library" to the other players in my group in case one of them perishes prematurely. These techniques will become prefabs when I'm making new characters.
Additionally, the campaign setting is Fantasy with a powerful group that is quite anti-magic. Innate magics and magic-like techniques are acceptable because the first can't be helped (the subject did not willingly "corrupt their soul" with the arcane) and the second is not technically magic, it's an immense amount of skill (That said, my first character is a Mage, so I'll be spending quite a bit of time on the run lol). These sword arts are intended to be high-skill techniques that can be attributed to skill.
And remember that I want to make attacks with the weapon instead of a generic HKA or RKA. You all have been doing well with that, but we've got multiple pages now so it may slip one's mind.
BACKLASH:
I do want to prevent using the teleport at-will. The ideal option would be something like "Teleport (Only to move behind a creature)" and "Trigger (When teleporting behind a creature; Make an attack against the creature, Uncontrollable, happens automatically, etc.)". You could use it to teleport to an ally (like in D:OS2), but you would still attack said ally (also like in D:OS2).
Would that work with a weapon attack? Sure you're swiftly moving the weapon through a hex to deal damage, but how is making a normal attack any different?
VANISHING BLADE:
You make a good point here. Perhaps a random move is in order as well...
I don't think I understand this sentiment... I want the invisibility to end after I make an attack. I get one attack while visible which makes me invisible. Then, I strike unexpectedly from somewhere else from invisibility and become visible. Picture most any fight scene that takes place in fog. You have that one character that the camera is focused on, on their guard and constantly looking around. Then you have the not visible but extremely haughty other character that usually is monologing about how superior they are. Then, they strike at the first character's back. Whether the attack is a success or not usually depends on the plot, but that's as far as this example needs to go.
Hmm... Yeah, if I go this route that would probably be a good idea.
If I did Invis-Desolid my GM would probably impose a +1 Advantage on the power due to what we call the "OP Combo". My Mage has a spell with Penetrative and Autofire that has this Advantage.
Hmm... Another good idea.
Flash could work... How about Flash plus Desolidify or Tunnel downwards a few meters instead of turning invisible? Could that work?
EXPLODING EDGE:
Duke Bushido, you mention Multipowers a few times. I do love Multipower Frameworks (genuinely. My Mage has 2. Of course, that's because my GM uses a specialized "Rule of X" so having a Multipower helps me keep my spells in check). However, I don't think that's the direction I should be going here. I feel like a martial art is better suited as you mentioned, but I don't really get how to use/make them. They seem both complicated and simple at the same time, which is usually a good indication that I have no clue what I'm doing.
... So... I had typed out another few hundred words explaining my intentions with the last two, but when moving quotes around I made a small mistake and hit CTRL + Z to undo it. It undid everything up to the beginning of me going on about Exploding Blade. It's really disheartening and I don't feel like spending the same amount of time typing it out again. The last two were pretty straightforwards, though. "Exploding" was a poor choice of words, however, because it was supposed to be a flat addition to damage (probably Normal damage) that would take a half-phase to set up. There wasn't actually going to be an explosion. I might ad a 2-hex cone AoE to it, however. My GM measures hexes as 2m each (but I read someone mention their hexes were 1m) so it would be a 4m cone.
Regardless, you all have had plenty of great input. I look forwards to what we can come up with for the first 2 techniques, as well as any other ideas we make here.