How "large" is a Lagrange point? When placing an object at a L-point, the natural assumption, as with all things gravity, is that you needn't place it in an exact spot to achieve a stable configuration. How much room do you have to play around with and still keep a stable configuration when placing an object at an L-point? As a consequence, can you place multiple objects at a single L-point? While having a stable position compared to the two-body system, would such objects at the same L-point also be "static" in relation to one another?
 A: 
How “large” is a Lagrange point?

L1, L2 and L3 are essentially zero size cause they're never stable.  They're still useful cause an orbital near L1, L2 or L3 doesn't require a lot of energy to stay in that general area. so we can use L1, 2 or 3 for not quite stable orbits that don't require much energy adjustment.
L1 Halo orbits are used too, not sitting near the Lagrange point, but orbiting kind of around it, or "about it" is I think the correct term.   Also called Lissajous orbits.

Given different distances in this L1 orbit, you can have several satellites orbiting L1 at the same time.   But none of them are really stable, but with a satellite that they only plan to get a few years of use out of, the long term instability isn't so much an issue.

When placing an object at a L-point, the natural assumption, as with
  all things gravity, is that you needn't place it in an exact spot to
  achieve a stable configuration.

That can be true but only for L4 and L5.   Never L1, L2 or L3.

How much room do you have to play around with and still keep a stable
  configuration when placing an object at an L-point?

The size of the stable zone in L4 and L5 depends on the other objects in the solar system.    For example, Further from the Sun the L4 and L5 zones grow larger.   Jupiter has huge L4 and L5 zones.    Saturn's L4 and L5 stability zones are smaller cause they're affected by Jupiter's large mass, similarly Uranus & Neptune's L4 and L5 zones are perturbed by the inner planets and not as large as a straight Sun/Planet/L4/L5 calculation would suggest.   Precise calculation gets quite mathematical.

As a consequence, can you place multiple objects at a single L-point?
  While having a stable position compared to the two-body system, would
  such objects at the same L-point also be "static" in relation to one
  another?

If you're talking about placing objects like man made satellites or space stations, the gravitational attraction between two of those would be very small and almost certainly negligible.   Beyond that, this diagram of Jupiter's Trojans seems to answer your question.   There appears to be insufficient attractive force to gravitationally clump Trojan objects.    If there was, Jupiter wouldn't have a few hundred thousand objects 
in L4 and L5.  (Source)
 
As to the size of Jupiter's stable L4 an L5, Wiki says there's a range of +/- .15 AU and a range of 26 degrees of Jupiter's orbit.   That's enormous.   If you could see the stable L4 and L5 of Jupiter from Earth, it the moon stretches across about 1/2 degree of the sky.   26 degrees would stretch over about 1/7th of the sky, granted that would be lengh wise and it would be less narrow, but that's still a vast area.  Thousands of times the size of the planet for example.
I don't know how big Earth's stable L4 and L5 areas are, but likely quite a bit smaller than 26% of Earth's orbit.   The math is too hard for me.    Here's a site that talks about some of the mathematics of trojans if interested:  http://www.merlyn.demon.co.uk/gravity4.htm
Feel free to correct if I missed something.
