Are omens dropped as items
Most Omens are also regular items, yes.

From the first edition rulebook (sorry, second edition rulebook isn't available online anywhere).


  Most omen cards (except the Bite card) are also items you keep,
  and they’re used like other items.


And


  Some omens, like the Dog, aren’t items at all, so they can’t
  be dropped or traded. They’re companions that follow
  the explorer who has custody of them. (The Dog, Girl, and
  Madman all work this way.)


The second edition added the keyword "Companion" to The Dog, Girl, and Madman.

A Companion can't be dropped, traded, or stolen. But if it is lost, usually due to the owner being killed, then it stays where it is until another player enters the room with it; at which point it starts following the new player instead. 


  If you die and you have a companion (the Dog, the Girl, or the Madman), its token stays in the room where you died. Set aside that omen card. If another explorer enters the room, he or she gains custody of that companion (and takes that omen card).