No. The sphere will remain on the same square until the wizard spends a move action to move it again, or the wizard moves out of the spell's range.
I quote your quote:
The sphere moves as long as you actively direct it (a move action for you); otherwise, it merely stays at rest and burns.
That means that while the wizard is inside the spell's range, AND the spell's duration has not expired, the sphere is active regardless if the wizard moves it or not. So if it occupies the same square with a target, and the target does not move out of the square, she will be burned by the sphere, on the wizards next round WITHOUT any action needed by the wizard.
-EDIT-
Flaming sphere's diameter is 5 ft, both in Pathfinder and 3.5.
So it is logical that a square occupied by the sphere is on fire for the duration of the spell, and anybody that is on the same square or passes through it is affected by the spell (Entitled to a reflex save or damage) regardless if the caster moves the sphere on her round. Credit to @Can Canbeck and @medivh for the comments clarifications :)
1. Radius Patterns follow Grid Lines
The ring version of Wall of Fire says this about its area of effect:
a ring of fire with a radius of up to 5 ft. per two levels
If you turn to the back of the DMG (p. 307) or look at this answer, you can see some patterns for radius spread effects. All of them spread along square edges. They're a ring of squares, rather than a circle. In this pattern, a square can't be half in or half out, because it only travels along the lines.
(This is actually also true for line effects, as PHB p. 176 shows. If a line goes through any part of a square, the entire square is affected. For something like Wall of Fire, an entire square has to be in one state or the other to follow with this, but if you place it along the grid lines it's fairly easy to do that.)
2. Yes
As you noted, Blade Barrier doesn't give a thickness or anything to imply that it's more than a curtain of blades (in fact, it says its a curtain of blades). You can treat it the same way as Wall of Fire: draw it on the grid lines. (The rules for area of spells mention that you should draw it starting from a grid intersection, so you can follow the grid lines.)
Regardless of the shape of the area, you select the point where the
spell originates, but otherwise you don’t control which creatures or
objects the spell affects. The point of origin of a spell is always a
grid intersection. When determining whether a given creature is within
the area of a spell, count out the distance from the point of origin
in squares just as you do when moving a character or when determining
the range for a ranged attack. The only difference is that instead of
counting from the center of one square to the center of the next, you
count from intersection to intersection.
This also makes the cover part of it easy to handle: anything on one side of that grid line attacking the other side has to deal with the cover portion of the spell.
3. Creature Size Matters
If you're placing it on grid lines (as you should), then yes, you can't actually place it on top of a medium creature. Large (or bigger) creatures take up multiple squares and you could have the barrier cast such that it appears in the middle of one. That makes this part of the spell make sense:
If you evoke the barrier so that it appears where creatures are, each
creature takes damage as if passing through the wall. Each such
creature can avoid the wall (ending up on the side of its choice) and
thus take no damage by making a successful Reflex save.
Medium or smaller creatures wouldn't be damaged, as the barrier is not "where the creatures are".
4. Because the spell says so
This point has no rules-as-written answer, except because the spell says so. :)
I'd speculate that the game designers did that with the reasoning that as the spell comes into existence, someone can react and get out of the way before it can fully attack. Whereas once it's already there, someone has to go through it. But I don't know of any rules, anywhere, that explain why they did it this way.
5. Yes, you can attack
Blade Barrier says this:
Any creature passing through the wall takes 1d6 points of damage per
caster level (maximum 15d6), with a Reflex save for half damage.
By a strict reading, passing through the wall is movement. By that reading: yes, you can attack through it without taking damage. I'm not aware of any rules clarification on that point.
Anybody attacking through it does have to deal with cover:
A blade barrier provides cover (+4 bonus to AC, +2 bonus on Reflex
saves) against attacks made through it.
Best Answer
I advise conferring with the GM about how they would adjudicate such a sphere catching multiple creatures before spending too many resources on it (which may require casting the spell in a mock combat situation with your party).
If you're the GM, decide how it works and be consistent. It's fine if other spells (or other force-effects, even Forcecage) work differently, but make sure that Resilient Sphere always works the same (barring specific situational modifiers, of course).
Were I GMing, I would give non-targets who would be caught a ref save to dive out of the way, in the direction of their choosing (using up movement from their next action); an acrobatics check (or saving by 5+/hitting a nat-20 on the check) would let them end the movement standing, else they'd be prone. Large (or larger) creatures who are partially in the space would get the ref save to choose whether to end up inside or outside of the sphere; failure would put them on whichever side the caster wants them to be on.