Are salaries from tax-exempts taxed the same as regular incomes?
For employees? Yes. Exactly the same. The only differences in the IRS employee tax regs for non-profits have to do with retirement plans and benefits fairness testing.
BTW it is more accurate to call them non-profits instead of tax exempts. No institution is completely tax exempt. They all are subject to UBIT (unrelated business income tax). They all have to pay social security taxes, unemployment taxes, and workman's comp taxes for their employees. And most of them have to pay certain kinds of property taxes. I worked for a state university back in the 70's that had to pay thousands every year in property tax for their typewriters. The revenue was earmarked for a local school district and the state wouldn't let them out of paying it, even though they never paid a dime in tax for their real property.
Yes, but the principals can still avoid taxes — by claiming very nice expenses from the non-profits for their activities.