Bounding Spheres probably would help you with many moving objects; you calculate the radius squared of the object, and track it from it's center. If the squared distance between the centers of two objects is less than the sum of the squared radii of the two objects, then you have a potential collision. Everything done with squared distances; no square roots.
You can sort nearest neighbors by the minimum squared distance between your objects. Collision detection can get complex, of course, and with non-spherically shaped objects, Bounding Spheres won't necessarily get you collision information, but it can prune your tree of objects you need to compare for collision quite nicely.
You'll need, of course, to track the CENTER of your object; and ideally you'd like for each object to be rigid, to avoid having to recalculate bounding sphere sizes (although the recalculation isn't particularly tough, particularly if you use a tree of rigid objects each with their own bounding sphere for the objects which are non-rigid; but it gets complicated).
Basically, to answer your question about data structures, you can keep all of your objects in a master array; I'd have a set of "region" arrays which consist of references to the objects in the master array that you can sort the objects into quickly based upon their cartesian coordinates; the "region' arrays should have an overlap defined of at least 2x the largest object radius in your master object array (if that's feasible; a question of object bounding sphere scaling vs. number of objects obviously comes up).
Once you've got that, you can do a quick collision test by comparing distances of all objects within a region to each other; again, this is where region definition becomes important, because you're doing a significant tradeoff of number of regions to number of comparisons. However, it's a little simpler just because your distance comparisons come down to simple subtractions (and abs() operations, of course).
Of course, then you have to do actual collision detection between your non-spherical objects, and that can be non-trivial, but you've reduced the number of potential comparisons very dramatically at that point.
Basically, it's a two-tiered system; the first is the region array, whereby you do a rough sort on your scene. Second, you have your intra-region distance comparison; wherein you're going to do your basic collision detection and collision flagging on the objects which have collided.
There probably is room in this sort of algorithm for use of trees in dynamic region determination to even out your region sizes to assure that your region size doesn't grow too rapidly with "crowded" regions; that sort of thing, though, is non-trivial, because with objects of differing sizes, your sort on density becomes... complex, to say the least.
