Yes, though they're hard to use...
Situation 1: When hidden, creatures from which you are hidden must guess what square you are in. This means that you do not know which square a creature is in, nor when the creature moves. In this example Attacky doesn't know that Heidie is still adjacent to her as soon as she becomes hidden. When Heidie moves away, Attacky still doesn't know which square Heidie is in, nor that Heidie has moved. Therefore, the triggering condition "attack when Heidie moves from the square" cannot be triggered until Attacky senses (sees, hears, etc.) Heidie moving.
Situation 2: This one is a little bit more complicated, depending on what the power actually is. The definition of Invisible states:
If a creature is invisible, it has several advantages against creatures that can’t see it: It has total concealment against them, it doesn’t provoke opportunity attacks from them, and they grant combat advantage to it.
The key here is that invisible creatures don't provoke opportunity attacks, however nothing is stated about immediate or free actions with triggers. In this example, it sounds like
Trigger: An adjacent enemy marked by you shifts
Effect: You shift 1 square.
In this case, the trigger doesn't specify that you need to see the marked enemy, only that it shifts. A strict interpretation is that, even if you don't know which square the marked enemy is in, so long as it shifts while being adjacent, you can shift 1 square. Of course, Attacky has no idea where Heidi shifted to, so would have to guess which way to shift.
Amusing follow-on to this, if Attacky had
Benefit: When you use your blurred step, you can teleport to any square adjacent to the triggering enemy instead of shifting.
Attacky would have to pick any square on the map and try to teleport to that square, and if it weren't adjacent to the triggering enemy (Attacky has no idea which square Heidi is in after the shift) the teleport would simply fail.
The Problem: The problem here is that Attacky still doesn't know that Heidi has shifted and Blurred Step is a power you have to actively use, rather than a feature that automatically activates. This is where the problem becomes one of meta-game: if both Heidi and Attacky are players, Attacky knows Heidi shifts from when the player says "I shift" and can react. If Heidi is controlled by the DM then the DM does not have to announce his or her intentions to shift Heidi and Attacky doesn't know when to use Blurred Step. I suppose, in theory, Attacky could keep shouting "I use Blurred Step!" until it activates.
On the other hand, if Attacky had a feat that said "any time a creature adjacent to you shifts, you shift 1 square" then Attacky would suddenly and unexpectedly feel the urge to shift one square and not really know why.