There is quite a lot of active research in computer science about this topic. Here is a recent survey: https://arxiv.org/abs/2007.01795
PAV has already been mentioned, and is a very good rule if you want to ensure proportional representation. Its sequential version is not known to guarantee any representation properties, though in practice it might do well. There are some other good rules around, including Phragmén’s rule, which unlike SeqPAV provably satisfies representation guarantees. Like SeqPAV, it produces outcomes that can easily be explained.
Regarding your question (2), the proportional rules I mentioned may in the worst case return a committee whose utility (i.e. utilitarian social welfare) is a 1/sqrt(k) fraction of the highest possible utility, where k is the size of the committee. See Section 5.5.2 of the survey I linked.