No, forced movement is not required to take place. It is also allowable to cause the creature to move less than the indicated number of squares, so long as a destination is not specified.
Pg 212 of the Rules Compendium reads as follows:
When a distance is specified, it is a maximum; the creature or effect producing the forced movement can move its target up to that number of squares (or none at all). For instance, a character's power might say, "You slide the target 4 squares (or "up to 4 squares"); both mean the character can move the target up to 4 squares or not move it at all.
When a destination is specified, it is absolute; the creature or effect must either move the target to that destination or not move it at all.
To put it another way, you could think of it as causing forced movement, but simply using that movement to move the target 0 squares.
There is no difference between pulls, pushes and slides in regards to not enacting forced movement, though obviously you cannot push or slide someone into blocking terrain such as a wall.
There is no difference in regards to source or type of the effect - again, think of it as moving them 0 squares.
You can choose to move the target less than the amount specified, so long as the description does not specify destination the target must reach. If the target cannot reach that destination with the distance specified, the forced movement fails altogether.
Yes, you can choose which targets you wish to move and which you do not in an area effect.
Yes, both those pushes are valid.
When you push a creature, each square you move it must place it farther away from you. [RC 211)
Now, 4e measurements use taxicab geometry, or Chebyshev distance, so the effect is... often nonintuitive. I'm pretty sure 4e floors are constantly shifting hyperplanes. My players have developed the mantra, "Circles are squares, squares are circles, and never draw a triangle on the grid!"
How to tell if your push is valid:
To tell if a creature is moving closer, further, or the same distance, count the number of squares the creature must enter to be adjacent to you if he starts in the first square, or the second. The more squares he must traverse to reach you, the further he is from you.
You also need to have line of effect to every square you push him into. He can't enter blocking terrain, and ignores difficult terrain.
The flavor text describing a power's visual appearance has no effect on the power's mechanical application.
Weird.
This actually means that the squares you can legally push a creature into differ, depending on if the creature is head-on with your square or adjacent by corners, because any square adjacent to both you and him is invalid for a push when he is also adjacent to you.
However, so long as you fulfill the above prerequisites, you're free to bounce the guy around zig-zag-style; nothing says the movement needs to describe a straight line (and given 4e geometry, I'm not sure what a straight line would look like anyway).
For completion's sake, pulls and slides:
For pulls, each square must be closer to you than the last, and slides don't care what your position is related to the target.
Best Answer
Dissonant Whispers kind of does this already
Having recently played a bard in an extended campaign, I got a lot of mileage out of Dissonant Whispers, which uniquely forces movement that provokes opportunity attacks:
It will allow for crazy combos
Most of the time, the damage output won't be increased that much, because most creatures only have one reaction. However, it's possible to set up very damaging combos against single targets.
For example, imagine an enemy surrounded by a paladin, a warlock, and an eldritch knight. On the warlock's turn, he casts eldritch blast and hits, allowing him to push his target 10ft without a save. The paladin and eldritch knight both get opportunity attacks: the paladin gets to smite, and the eldritch knight, with war caster, gets to cast a spell. Moreover, because the warlock can shoot multiple eldritch blasts, this combo could theoretically be triggered multiple times in one turn if there are enough friendly creatures to make OAs. You can easily see how powerful such a combo can be.
Of course, Dissonant Whispers means that these combos are possible in vanilla 5e, but your tweak allows classes like warlocks to do it for free and without a save. Whether or not that breaks your game will depend a lot on how your PCs and monsters are built, but it opens the possibility of very powerful single-target combo attacks.
I realize that these combos are pretty much the intent of this houserule, but my point is that a party that builds around this houserule will become far more powerful than a party that doesn't.