Algorithm to select sets of objects while maximizing number of objects covered
The problem you're describing can be seen as the weighted independent set problem, as follows: Construct a graph $G=(V,E)$, where every node $v$ in $V$ corresponds to your sets of objects. The nodes $v,w$ have an edge in $E$ if and only if they share an object. The weight of each node is the number of objects contained within the corresponding set. 

Now, observe that maximizing the amount of objects used corresponds to finding a set of nodes in $G$ such that 1. none of the nodes are adjacent and 2. the sum of weights over these nodes is maximized. This is precisely the weighted independent set problem.

Unfortunately, this problem is NP-hard, which means an exact efficient algorithm is unlikely to exist (and even less likely to be found). This means that, unless you only have to deal with small values, you would either have to resort to heuristic or approximation algorithms, of which there are a few in the literature. It may be possible that only a particular kind of graphs arise (e.g. trees). In that case, there may be an efficient algorithm, but that depends on your input.