First, can I use Shadow Jaunt with a move action to appear next to an enemy?
No: shadow jaunt is a standard action. The move-action teleport is shadow stride.
First, can I use Shadow [Stride] with a move action to appear next to an enemy?
Yes; why wouldn’t you be able to? Nothing in the maneuver mentions any such limitation.
Next, can I use a standard action to attack or use a strike maneuver?
Yes; why wouldn’t you be able to? Nothing in the maneuver mentions any such limitation.
Last, can I use Shadow Stride with a swift action to move away?
No: shadow stride is a move action. The swift-action teleport is shadow blink.
Last, can I use Shadow [Blink] with a swift action to move away?
Yes; why wouldn’t you be able to? Nothing in the maneuver mentions any such limitation.
All in one turn?
Yes; why wouldn’t you be able to? Nothing in any of the maneuver mentions any limitation on combining these.
Does teleporting away with either the Jaunt or Stride offer an AoO?
This is actually unclear. “Moving out of a threatened square usually provokes an attack of opportunity from the threatening opponent.” Does teleportation count as “moving?” Unclear. Most tables seem to play that teleportation does not count, does not provoke.
Can I use the tumble skill to avoid it?
Probably not; that specifies “normal movement.”
Would that cause the teleportation distance to halve?
I guess? We’re kind of getting into ruling/houserule territory.