Probably shouldn't tell border patrol you're doing unpaid labor in a country you're visiting. When speaking to authorities it's best to say the absolute minimum required for the encounter.
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If there is ambiguity, then we can't have them here.
Generally yes.
But you can have on-campus jobs to supplement your income, and there are at least two programs (CPT and OPT) that let you get approval for limited-term employment in your area of study. CPT also requires university approval.