5 is really only debated by academics. I think the actions of policy makers prove they think it's true. Does anyone remember the debate during the Clinton years over whether or not they should start paying back government debt given a surplus?
Item 5 is theoretically true if you assume accounting rules hold. The sectoral balances formula (derived from the definition of GDP) doesn't hold otherwise.
The only way for the economy to grow if the Government is running surpluses is with increasing private debt (so not really "paying down" anything but just transferring debt from one sector of the economy to another), and/or with a big enough trade surplus to cancel out the other sectors.
