A Halved Word™ is a word that,
when doubled and then anagrammed, produces another English word. As f'' pointed out, each sentence contains a reference to the larger word.
Scene
scenescene -> senescence (declination dodue to aging)
Agrin
agrinagrin -> arraigning (brought people before the court)
Horse
horsehorse -> horseshoer (the man who gave him new shoes)
Icons
iconsicons -> concisions (much information in a tiny space)
Stein
steinstein -> intestines (human entrails)
Metro
metrometro -> tromometer (thanks, f'') (device to detect tiny earthquakes)
Latte
lattelatte -> tattletale (snitched)
Host
hosthost -> hotshots (highly successful, aggressive people)
Pear
pearpear -> reappear (come back later)
Four
fourfour -> froufrou (women in highly decorated dresses)