Can a runner trash ICE or Operation cards after a run on the HQ?
No, Runners do not have the option of trashing cards that have no trash costs. They may trash cards with trash costs. Special care should be taken when accessing multiple cards (see page 18). From page 18 of the rules, under Access Phase:

HQ: The Runner accesses one random card from HQ and any upgrades in its root. Any cards the Runner does not score or trash return to HQ.
Trashing Cards - If the Runner accesses a card with a trash cost, he may pay credits equal to its trash cost in order to trash it to Archives faceup.