I think this hypothesis has sociological and psychological merit, and I'm sure its been tested before, though I don't know of any studies off-hand.
Obviously that's conjecture, and all poor people aren't criminals (and all criminals are not poor), but it's not a leap of logic to say poverty can lead normally straight people to mentally justify committing a crime. Welfare programs can and do help people get back on their feet.
Living in Michigan, almost all of my family became unemployed as a result of the factories closing down. Some took the assistance to get retraining and find another job, and some took maintaining the assistance as their new job or as a retirement package. Some just weren't able to get back into the job market no matter how they tried. I got out of my state's "No Worker Left Behind" retraining assistance just before the new governor shut down the incredibly successful program.
It's a combination of willingness to work and ability to work. Welfare programs can do no more than take basic survival off your "worries" list.
You (mostly) had me up to the first point.
How are food stamps a positive / negative on crime?