OK, I'll try my best at explaining.
The magnet on a speaker is magnetized vertically, so that the top is the north (I think) pole, and the bottom is the south. The top plate is attached to the top of the magnet, causing it to act as an extension of the northern pole. And the pole piece, because it is attached to the back plate, will act as an extension of the southern pole. So I assume that removing the pole piece would either cause the speaker to sound like crap, or maybe barely function.
The coil question is linked to the pole piece question in someway. In a very good speaker (lets say DD) the coil will have exactly the same amount of coil on either side of the gap. Now, lets say the coil is 2.5"s long, and the gap (the thickness of the top plate, pretty much) is 0.5"s. This means the the speaker can move linearly 1" in each direction, it can do more (maybe even double!) but it won't be as linear. To stay perfectly (or as perfect as we can make a speaker, which is pretty damn shocking) linear, the gap must have its full length, 0.5"in our example, full with coil.
Now, take the coil just out of the gap, not only will it sound shocking, the closest to linear it will get is at the bottom of its down stroke. With this placement, our example DD would have to be moving a total of 1" just for the gap to be full at the bottom of the stroke, up this to 2"s of excursion and the bottom 0.5"s will be the only part that is linear.
So unless the coil is completely centered (1" either side of the gap with our example DD) it will behave very badly and have very bad sounding and quiet output. If the spider has sagged 0.25" you will probably not ,realistically, notice a difference, but if it is as bad as your first or third examples, it will suck.
As for how far is too far. I'm sure you could get some vibration with the coil 12"s away from the gap. The magnets are quite strong, go round to grannies and sit your DD lvl5 on the table and I'm sure she will be asking WTF is happening with the picture on her 20 year old tv (which is 6-10ft away from the table).