I know that we already have satellites in position at our Lagrange points, but what if we want to use them to park spacecraft sections for assembly reasons, or possibly even a meteorite for mining.
Are the 10 points big/stable enough to have multiple objects stationed there without causing collisions? If not for all 10, which ones?
Do we already have points that have more than one objects?
When the media talks about spacecraft at the Lagrange point, what they really are saying is that the spacecraft will be somewhere in the general area of the Lagrange point, not at the point itself which is infinitely small.
The gravitational pull of one of the bodies will be a bit higher, but in practical term it does not make any difference, so yes, the Lagrange "points" are big enough.
Are they stable? Again, for practical purposes they are i.e. with some modest station keeping you can keep a spacecraft there.
More details can be found here:
Lagrangian points article [astronomy.com]
Lagrangian points - stability [wikipedia]
I presume you are referring to the five Lagrange points for the earth and the sun (SE) and the five points for the earth and the moon (EM), to get to your tally of ten points.
Broadly speaking for systems like SE and EM, L1-3 lie in a straight line between the bodies and are unstable. L4 and L5 are pretty stable when the larger body is ~>25 times the size of the smaller one. This is just about true for EM and is obviously true for the earth and sun. Practically speaking l4 and l5 are generally not as useful as the other Lagrange points in terms of sun earth observation, getting shielding from the sun or as a staging point for BEO exploration.
You may have heard of the three body problem, and this is the reason why there is no definitive long term stable orbit at L1-3.
There are some relatively predictable associated orbits called Lissajous orbits which don't solve the three body problem but can let you park a spacecraft in an orbit around any of the L1-3 points with some very minimal station keeping. These are the orbits that have been used to date for satellites, but are not very useful for storing a large asteroid unless we also add a large enough propulsion system to keep it in the Lissajous orbit.
