No it doesn't. Having to poach employees to increase your labor pool doesn't mean that you are willing to raise wages without and bound. A food crisis is where an inelastic demand meets a hard supply limit, so prices can go up dramatically, but they don't go up infinitely, not even in a famine.
Supply and demand does not allow for unbounded rises in prices because demand is never infinite. Rather, prices go up to the highest amount demand will allow for. For things like food for which demand is inelastic, prices can go up pretty high and people will eat into their savings if they have to, but labor will more likely be limited by the marginal productivity of individual workers.
Generally, yes, but it's not an absolute.
Each time an employee is poached for more money, that person has more money to spend; done frequently enough, this will increase the velocity of money and cause general inflation or even hyper-inflation.
I'm only speaking hypothetically because folks are talking about 0% unemployment.
If humans have choice, some will always choose to quit their jobs on any given day.
So the situation you're describing is nonsensical; it's incoherent to try to extrapolate what it would mean because the premises are incoherent.
This explains it better.
Milton Friedman's "natural rate" of unemployment is basically "the rate capitalists are gunning for" in an attempt to suppress wages. Quite a bit below full.
You can tell from ideas like this that Engels never had to work a day in his life. Not everyone is going to instantly switch jobs when they hear about a slightly better one. And I thought capitalists were the ones that supposedly treats humans as mindless automatons.
If you're going to assume spherical cows like that, all businesses would already make 0 profits due to competition. In a hypothetical capitalist world with 0 unemployment, employers wouldn't poach anyone unless if they think that it would be a profitable decision. We'd have higher inflation, but that's not really the end of the world if everyone is employed. More importantly, our economy would stagnate due to not having any slack, but that's the to occur in any non-capitalism system that has 0 unemployment.
Man, the idea of a labor-force participation rate being a tracked statistic by central banks would have blown Engel's mind.