Federal Taxes paid by Schools

What taxes do State , Territory and private schools pay to the federal Government,if any?
Do charities pay taxes?
Are all churches exempt from paying taxes?
I dont know about schools but, I work for a charity and we get all GST back. Also all employees are paid a special way whereas we are only taxed on a certain percentage of our wage. Doesnt benefit employees any but the organisation doesnt pay the government the tax they would be deducting from my wage. Its about $20k difference, so the organisation get to pay about $6K less tax to the gov on my behalf (wish I got it).

Actually in the long run, I lose out slightly as my super is based on the amount I do get taxed on = less super put in and I cant claim as much with neg gearing. Also centrelink treat me as though I have earned the full taxable amount so I get my child care % charged on the full amount.

Confusing I know.