Yes, but not in general shortages of doctors. It's a totally different problem, so this isn't really a helpful way to look at it. After all, if there's a shortage of cabbages, it doesn't mean the remaining cabbages benefit and can lobby to keep it that way!
Cabbage farmers can, though.
Yes, but when that sort of thing happens in commodities it's an illegal cartel, whereas when it happens with jobs it's responsible self-regulation by a professional association.