Since poverty is (at least typically) defined relative to some average those two things are not as disconnected as you may think.
But in general I'd expect crime to be a "least worst option" choice and I seem to remember research to that effect. Although I wouldn't be able to find that now again, so take it with an appropriate amount of salt.
No, they are orthogonal. A society split between middle class workers and billionaires still has inequality without poverty. A society of 100% poverty has no inequality.
"Two commonly used measures are:
people in relative low income – living in households with income below 60% of the median in that year;
people in absolute low income – living in households with income below 60% of (inflation-adjusted) median income in some base year, usually 2010/11."
This is however a bureaucratic definition of poverty that I don't accept as capturing the everyday meaning, but I thought it might be of interest to mention. I remember being under the poverty line and thinking "this doesn't count. I don't accept that I'm living in poverty." Knowing these technical uses made me regard official-sounding news and reports about 'poverty' quite differently.
(The bottom of the article talks about a group giving a more lay-compatible definition "The Social Metrics Commission (SMC) proposed a new based on the extent to which someone’s resources meet their needs. This accounts for differences among households such as costs of childcare and disability, savings and access to assets." which seems more meaningful if you're interested about absolute standards of living).
That's good to know. I had similar feelings about wealth. At one point my wife and I were in the bottom 10% or lower, because of our student debt. Yeah when we graduated medical and dental school we had pretty negative net worth. But... does it really make sense to claim we had no wealth? Or that someone who made 20k a year with no debt should trade places with us? I think wealth and inequality studies are important, and if they don't address this issue they are misleading, even meaningless statistics.