Some of the rules text regarding moveable zones is a little ambiguous. Wax Eagle has interpreted things one way, which I find counter-intuitive and strange. Here's another interpretation, which I believe is the intended one.
When considering these moveable zone rules, imagine a wizard shooting a cone of flame from his hands. This is a Close blast.
Then imagine the wizard walking or simply turning 180 degrees while continuing to shoot the cone of fire. This is a moveable zone originating from a close blast.
The fire will affect anything within a certain range of the wizard, so long as nothing is in the way. If the fire passes by a pillar, then anything behind the pillar will not be affected by the fire. This is blocking terrain temporarily blocking the zone's effect in a square that has no line of effect from the origin square.
Once the blast is past the pillar, that space will be re-filled with fire, as there is no longer anything in the way.
1) Did the creation of the zone get blocked by line of effect? Therefore the zones effects are not in effect behind the B.
The origin square of a blast is the square of the caster. The zone is created just fine, including the square behind B (note 1). However, there is no line of effect (RC 107) to the square behind B, so the zone's effects do not apply to this square (RC 109).
There is no line of effect because every line traced from the origin square to the blocked square "passes through or touches blocking terrain".
2) Does this change the zone if the caster C were to move down one, creating line of effect into the square just to the right of B?
It does not change the zone's size or shape, the zone is a 3x3 square. However, since there is now line of effect, the zone's effects now apply to this square.
Even though there is blocking terrain in the zone, it hasn't been moved through blocking terrain because you had line of effect to at least one square of the zone at all points (note 2). If a turn had ended with the caster not having line of effect to any squares in the zone, then the zone would end at that point (RC 121, PH 59). Hence, the zone cannot be moved through blocking terrain.
3) If the answer to 2 is no, would it change if the zone 'follows' the caster (in the case of a burst 3 that creates a zone)? Related, if the zone follows the caster, does it maintain its shape? (in the above case, it would stay shaped like a C).
Nothing is different if the zone started as a burst rather than a blast.
4) If the answer to 2 is yes, does the zone stay still, but the square's effected by it change?
If the zone is a moveable zone, then it moves when moved. If it is not a moveable zone, then the caster moving after it's already been cast doesn't affect it in any way.
5) Do square's that now have no line of effect get removed from the zone?
The zone's shape never changes. If the zone moves, and a square in the zone no longer has line of effect from the origin (the caster), then the zone's effects don't apply to that square.
6) If someone were to put blocking terrain in the middle of a zone, would the zone behind the blocking terrain cease to have an effect?
Yes, any squares that lose line of effect from the origin square would no longer be affected by the zone.
1) The Rules Compendium and the Player's Handbook disagree a little bit here. Much of the text in the Rules Compendium is inconsistent and unclear regarding zones. The PH says that "unless otherwise specified, a zone fills a power's area of effect". The RC says that "the zone fills each square in a specific area, which is usually a burst or blast" which is fine, but then goes on to say "the squares must be within line of effect of the origin square" which directly contradicts the previous sentence. I'm going with the most consistent interpretation, which agrees with the blast rules in that the square is part of the zone, but the effects don't apply to targets in that square.
To further support this position, consider these lines from the Burst and Blast sections in the RC:
Even if a creature is in the area of the fire, the creature isn't affected by the fire if it is is behind an obstacle that blocks line of effect between the creature and the origin square.
Even if a creature is in the area of Albanon's shock sphere, the creature isn't affected by the lightning if it is behind an obstacle that blocks line of effect between the creature and the origin square.
2) The Player's Handbook is very clear about this:
If a power allows you to move a zone, at least 1 square that the zone covers must remain within the power’s range. If you move far enough away from a zone that it is no longer in range, its effects immediately end.
But they chose to reword things for the Rules Compendium, and inadvertently introduced an ambiguity:
At the end of the creator's turn, a movable zone ends if the creator doesn't have line of effect to at least 1 square of the zone or if the creator isn't within range (using the power's range) of at least 1 square of the zone.
From the PH wording, it should be pretty clear that this is intended to be parsed as "doesn't have (line of effect to at least 1 square of the zone)" i.e. "doesn't have line of effect to any squares in the zone", not "doesn't have line of effect to (at least 1 square of the zone)" i.e. "there exists at least one square of the zone to which the creator doesn't have line of effect".