Will things like electric shavers, hair dryers etc. from UK work
straight from the mains (using the 2 flat pin adapter) in USA? Does
one need a "step down" transformer / voltage converter?

It will be FAR cheaper to buy a new 110V hairdryer than to buy equipment to
change the voltage. You'd need a very hefty (expensive and heavy)
transformer to convert that sort of power. The shaver may work straight
away. Check the rating panel to see if the voltage range covers 110V AC
(60Hz). If not, buy a new one in the US.

Christian.