reuse padding when installing new carpet
To me, this seems like a "penny-wise, pound-foolish" place to save a few pennies. But then, I tend to rip out carpets and replace them with hard surfaces. 

Consider that whatever spills, grunge or general nastiness that got through the old carpet are in the pad - and consider that the pad is already somewhat compressed in the wear areas. You're giving away a good deal of the "new-carpet experience" by leaving a dirty old half-flat pad under it. 

Tack strips I don't feel as strongly about reusing.